Don't understand MOSFET as amplifier

It is an amplifier because you put a wiggly signal in, and you get a bigger wiggly signal out. The fact that the output wiggles down when the input wiggles up is a trivial detail, that -- if it's a problem at all -- can be solved in any number of ways (not least of which is following one stage by another, because two negative gains, when multiplied, result in a positive gain).

You need to distinguish between large signal and small signal.

Suppose your input is a DC value VIN. This causes a DC current ID which is calculated with your equation.

Now, suppose you add a small signal to this: Then you can use Taylor's expansion to get an approximation, which will yield:

iD = (what you have) + (derivative, evaluated at the point you have)*input.

What this means is for oscillations around a DC value in the input, you get oscillations around a DC Value in the output, and both have the same frequency.

The relationship between the oscillation term in the output and the one in the input is called gain, and sometimes is negative, but that only means the output is a mirrored image of the output. What matters is: if the absolute value of the gain is greater than 1, the output produces a "larger" version of the input.