How does a preamplifier improve the Signal-to-Noise Ratio (SNR)?
There's two things to consider with a preamp which explains why it improves SNR.

The first, and most important, is that the pre-amp is closer to the source (such as a microphone).  There's fewer noise-generating components between the microphone and the pre-amp than there would be if you waited all the way to the main amplifier.  For example, quite often there are treble/bass knobs between the pre-amp an the amplifier to adjust the tone of the sound.  These devices add noise, but because they occur after the pre-amp, they're adding noise to a 1V signal from a low-impedance source instead of adding noise to the tiny signal that comes from the high-impedance microphone.

The second advantage is that it's easier to do clean amplification at low currents/voltages.  It's much easier to do the amplification in two steps: one step raises it from the tiny signal from the microphone, and the other takes that signal and powers the speakers.  In fact, it's likely that if you didn't have a pre-amp before your tone controls, you would likely have a 2 stage amplifier (which is a pre-amp and main amp slapped together).