LDR not detecting light from LED when finger placed between the two
The situation is simple. If you make it impossible for light to enter the LDR then the resistance of that device is increasing as you can see from the added picture.


Removing the light with your finger makes the resistance go up. The result is that the base of Q1 does not receive enough current anymore to make it conduct. The collector of Q1 goes up and Q2 starts to conduct in full. So the led D! starts to glow.

The next step is to understand what happens when the led is putting it's light on the LDR.

Looking at the schematic again you can see that if Q1 starts to conduct complete  Q2 will be cut of and the led goes off. However that is not possible because then there is no light falling on the LDR anymore.

The result will be a balance whereby the resistance of the LDR lowers enough with the light from the LED to keep Q1 and Q2 conducting so much that the light from the led is sufficiant to keep the LDR at the resistance needed.

Now if you start playing with this combination then you are able to change the balance and send the resulting signal to the analog input of arduino.