Understanding influence and using it properly
First of all, are you confusing influence and thickness? See my answer here for the difference.

Influence and thickness obviously have to lead to territory in the end, the questions is, how?

The rationale behind "Do not use thickness to make territory" is to not use it directly in a crude way. Imagine a situation like this:

$$Wcm1 Black's direction of attack is bad
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . X O . . . . . |
$$ | . . . . . . 1 . 2 . . . X O . O O O . |
$$ | . . X , . . . . . , . . X X X X X O . |
$$ | . . . . . . . . . . . . . . . . . X . |
$$ | . . . . . . . . . . . . . . . . X . . |
$$ | . . . . . . . . . . . . . . . . . . . |


The extension to B2 is poor direction, as it, while creating some territory, is not using the wall to the fullest. White can think about attaching to the corner stone or simply extending to gain at least part of a base, and black will have a hard time attacking. Compare to this:

$$Wcm1 Right direction
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . X O . . . . . |
$$ | . . . . 2 . 1 . . . . . X O . O O O . |
$$ | . . X , . . . . . , . . X X X X X O . |
$$ | . . . . . . 3 . . . . . . . . . . X . |
$$ | . . . . . . . . . . . . . . . . X . . |
$$ | . . . . . . . . . . . . . . . . . . . |


White does not have any base and is even forced to jump to the center, not having a single eye. Black can look forward to keep attacking. At the same time, black made some territory in the upper left.

$$Wcm1 Extending is difficult
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . X O . . . . . |
$$ | . . . . 2 . 1 . . 3 . . X O . O O O . |
$$ | . . X , . . . . . , . . X X X X X O . |
$$ | . . . . . . 4 . . 5 . . . . . . . X . |
$$ | . . . . . . . . . . . . . . . . X . . |
$$ | . . . . . . . . . 6 . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |


Using thickness to attack

The word thickness, as used in Go, is approximately synonymous with strength A group of stones is thick if it contains few or no weak points and is not vulnerable to attack. It should be obvious that thickness is a good thing to have. Your opponent must be extremely cautious in the vicinity of your thickness, for he will not find any useful kikashi to play against it should his stones get into trouble. Backed up by thickness, you can attack with carefree abandon. Also, in a ko fight the player whose groups are thick will have the advantage. Hence, in the first half of the game, building thickness is just as important as making territory. The key idea in handling thickness is to treat it with respect and not to play too close to it, whether it be your opponent’s thickness or your own.

Let's consider another example.

$$Wcm1 White is very thick but horribly abuses the value
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . X . , . . . |
$$ | . . O . . . . . . . X . X . . . . . . |
$$ | . . . . . . . . . . . . O X . . . . . |
$$ | . . . . . . . . . . . . O X . X . . . |
$$ | . . . . . . . . . . . O . O X . . . . |
$$ | . . X . . . . . . 2 . . O O O X . . . |
$$ | . . . , . . . . . , . . . . O , . . . |
$$ | . . . X . . . . . X . 1 . . . O . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ +---------------------------------------+


The white extension to 1 in this example is, well, hideous. Clearly Black 2 is a larger move than White 1, fixing black's problems and creating territory.

Instead, white should invade like this:

$$Wcm1 Use the thickness to attack
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . X . , . . . |
$$ | . . O . . . . . . . X . X . . . . . . |
$$ | . . . . . . . . . . . . O X . . . . . |
$$ | . . . . . 0 . . . . . . O X . X . . . |
$$ | . . . . 8 9 . . . . . O . O X . . . . |
$$ | . . X . 6 7 . . . 2 . . O O O X . . . |
$$ | . . . 4 5 . . . . , . . . . O , . . . |
$$ | . . . X 3 . . 1 . X . . . . . O . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ +---------------------------------------+


Continuation

$$Bcm1 Use the thickness to attack
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . X . , . . . |
$$ | . . O . . . . . . . X . X . . . . . . |
$$ | . . . . . . . . . . . . O X . . . . . |
$$ | . . . . . 1 2 . . 9 . . O X . X . . . |
$$ | . . . . X O . . . . . O . O X . . . . |
$$ | . . X . X O . 4 . X . . O O O X . . . |
$$ | . . . X O 3 . . . , . . . . O , . . . |
$$ | . . . X O 5 . O . X . a . . . O . . . |
$$ | . . . . 6 7 8 . . . . . . . . . . . . |
$$ | . . . . . 0 . . . . . . . . . . . . . |
$$ +---------------------------------------+


Black has some difficulty finding severe moves. Due to the presence of White’s thickness he had to make defensive plays at 2 (dia1) and 9 (dia2), and the invasion was a success for White. Note that 'a' still remains as a large yose point for her.

$$Bcm1 Black can hardly fight back
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . X . , . . . |
$$ | . . O . . . . . . . X . X . . . . . . |
$$ | . . . . . . . . . . . . O X . . . . . |
$$ | . . . . . X O . . . . . O X . X . . . |
$$ | . . . . X O . . 2 . . O . O X . . . . |
$$ | . . X . X O . O . X . . O O O X . . . |
$$ | . . . X O X . . . , . . . . O , . . . |
$$ | . . . X O X . O . X . . 3 . . O . . . |
$$ | . . . 1 O X O . . . 4 . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ +---------------------------------------+


If Black had used 18 in the last diagram to capture three stones, as with 1 in this diagram, he would have suffered a big loss. White 2 captures a large territory. It is pointless for Black to extend into the teeth of White’s thickness with 3. White 4, in conjunction with White’s thickness, deprives Black of eyes.

When your opponent has made thickness, you should try to deal with it by establishing some stones a safe distance away from it. After making such a base, you can approach more closely. However, slowly approaching thickness is, well, slow - it means you won't make points as quickly. This is a drawback of allowing your opponent to have thickness.

Creating a moyo

$$Wcm1 Nice moyo for black, thanks to the newly created thickness.
$$ | . . . . . . . . . . . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . 2 . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . 8 1 . . |
$$ | . . X . . . . . . . . . . . . 6 7 . . |
$$ | . . . , . . . . . # . . . . . X 5 . . |
$$ | . . . X . . . . . . . . . . . 4 3 . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ +---------------------------------------+


In response to White 1, the niken takabasami of Black 2 in conjunction with Black , is a good strategy. The 3-3 invasion of the corner with White 3 is the usual continuation and with the sequence to 8, Black has made thickness on the outside while White has taken profit in the corner. The marked stone has now become an ideally placed as it is a good extension from both the shimari in the upper right corner and the wall on the lower right side.

Overconcentration

You probably already know this situation:

$$Bcm1 Note that the white extension is 2 space, not 3 space at 'a'.
$$ | . . . . . . . . . . .
$$ | . . . X . . . . . . .
$$ | . . . . . . . . . . .
$$ | . . . X . 2 . . . . .
$$ | . . . . 1 O . . O a .
$$ | . . . . . . . . . . .
$$ | . . . . . . . . . . .
$$ +----------------------


By kicking at 1, black can force overconcentration of the white group.

The very same idea can be used to counter bad use of thickness. I've found a great example for this:

$$Wcm1 Imagine this situation.
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . X . . . . . , . . . . . X . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . X . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . O . . |
$$ | . . X , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . O . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . X . . . . |
$$ | . . . . . . . . . . . . . X . . X . . |
$$ | . . . X . . . . . , . . O O X X O . . |
$$ | . . . . . 1 . . . . . . O X O O X X . |
$$ | . . . . . . . . . . . . O X O . O . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ +---------------------------------------+


White is trying to use her thickness to create territory on the lower edge. We can make use of this wrong attempt.

$$
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . 7 . . . . . . . . . . X . . . . |
$$ | . . . . . 4 . . . . . . . X . . X . . |
$$ | . . . X 5 1 2 . . , . . O O X X O . . |
$$ | . . . . 3 O 6 . . . . . O X O O X X . |
$$ | . . . . . . . . . . . . O X O . O . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ +---------------------------------------+


Black should play tsuke-osae with 1 and 3. The sequence to Black 7, which is joseki, will follow, leaving Black with a large profit in the corner and White with thickness along the lower side. But what is this thickness worth?

$$
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . 7 . . . . . . . . . . X . . . . |
$$ | . . . . . O . . . . . . . X . . X . . |
$$ | . . . X X X O . b , a . O O X X O . . |
$$ | . . . . X O O . . . . . O X O O X X . |
$$ | . . . . . . . . . . . . O X O . O . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ +---------------------------------------+


If White were to make an extension from his formation on the lower left side, the ideal
point would be at 'a'. On the other hand, the ideal point for an extension from the
formation on the lower right side would be at 'b'. The sphere of influence of these two
formations overlap. Eventually, White may have to play on one of these points to prevent Black from invading. But, for White to play on one of the points will result in overconcentration.