Squeezing of laser light generally involves a non-linear interaction, where the nature of the interaction depends on the intensity of the light that is present. An easy to understand example is frequency doubling, which takes two photons from a pump laser, and sends out one photon of twice the frequency.
You can think of the input beam as a stream of photons with some fluctuation in the "spacing" of the photons along the beam. That is, on average you will receive, say, one photon per some unit of time, but sometimes you get two, and sometimes none.
If you send this beam into a nonlinear crystal to do frequency doubling, the doubling will occur only in those instants when you get two photons in one unit of time. In that case, the two photons are removed from the original beam, and produce one photon in the frequency-doubled output beam.
If you look at the transmitted light left behind in the input beam, you will find lower fluctuations in the intensity, because all of the two-photon instants have been removed. Thus, the transmitted beam is "amplitude-squeezed." It's not quite as obvious that the frequency-doubled beam has lower intensity fluctuations, but it, too is amplitude squeezed, because you get photons only at the times when you had two photons in the original beam, and it's exceedingly unlikely that you would get two of those in very close succession (or four photons from the original beam in one instant). So you have a lower intensity in the doubled beam, and also lower fluctuations.
So, for example, your input beam might give the following sequence of photon numbers in one-unit time steps:
The input beam after the doubling crystal will look like:
and the doubled output beam will look like:
Both of those have lower fluctuations than the initial state.