Why does the small signal analysis work? (Intuition)
In small signal-analysis, the behavior of a non-linear device is approximated as linear about a DC operating point (Quiescent point). Basically we put in a small 'wiggle' signal that doesn't change things to much from the DC level. With a linear model we can apply the principle of superposition. 

That is, the total response of the system can be viewed as the sum of the responses to each source individually. In your left picture there are two sources, a signal input from the left, and what is usually a DC power supply at the top. In the right hand picture, this source is simply being zeroed so the response to the input signal can be more easily determined. Therefore it is replaced by a short circuit to ground.

The actual behavior of the circuit is then this response, plus the response to the DC source alone.