What's the point of putting five face-up train cards next to the deck in Ticket to Ride?
When you are taking cards on your turn, you can choose to take any of the face-up cards rather than taking one from the deck. Taking a face-up card this way lets you choose what you are getting (as long as the color you want is available).

From the rules:


  Draw Train Car Cards – The player may draw 2 Train Car cards. He may take any one of the face-up cards or he may draw the top card from the deck (this is a blind draw). If he draws a face up card, he immediately turns a replacement card face-up from the deck. He then draws his second card, either from the face up cards or from the top of the deck. (See Train Car Cards for special rules for Locomotive cards).


This is a key part of the game, so I recommend reading through the rulebook again; if you missed that you likely missed other important things.

One rule that's related (though not all versions have this, Nordic Countries doesn't):


  If a Locomotive card is one of the five face-up cards, the player who draws it may only draw one card, instead of two. If, after having drawn one card the replacement card is a Locomotive, the player cannot take it. If at any time, three of the five face-up cards are Locomotives, all five cards are immediately discarded and five new ones are turned face-up to replace them.


To summarize; if you are choosing to draw 2 train cards on your turn; you choose one at a time to draw the top card of the deck, or one of the 5 face-up cards. If you chose a face-up card, you replace it with one from the top of the deck, and then you make the choice a second time. If you choose to take a locomotive card from the face-up cards, then you can only take 1 card that turn instead of 2.