The input gain control boosts or cuts a signal to a usable level for the mixer. Where it is true that the police won't arrest you if you have this set wrong, having it set correctly is ideal.
Think of it as a car's tire pressure. The ideal pressure for everyday driving is usually around 32psi with all 4 tires at the same pressure. Sure your car will still drive with 10psi in one tire and 40psi in the rest, but things like handling, fuel milage, and tire life would be affected.
I'm a firm believe of being experimental when it comes to mixing, but there are certain things that you can do so you're not set up for failure right out of the gate. Proper gain stage gives you a good clean signal to use, and everything downstream (eq, compressor, reverb, delay etc) will have this ideal signal voltage and will function to their full capability. Low gain will usually result in a muddy, lifeless mix. High gain will over drive the input and distort your signal.
Signal to noise ratio: Any amplified signal is made of two parts, the desired signal (from your microphone diaphragm or guitar pickups, keyboard etc) and background noise caused by radio, electrical and magnetic interference either on the stage or from inside the mixer itself. The mixer will amplify 100% of the combined signal and doesn't care if the signal it receives is from your lead singer or the transformer in your power supply . Its your job as a sound engineer to feed the mixer input with the signal you want and filter out as much noise as possible. I'm going to steal some of an analogy used when explaining power factor, I'll try to apply it to signal to noise ratio as best as possible.
You are a bartender(sound engineer) and a customer just ordered a beer. The contents of a beer mug is beer(signal) and foam(noise). If the keg is too warm or the beer is poured too fast, you'll have mostly foam and very little beer. You're customer is not getting their money's worth and it won't quench their thirst. If it's poured properly at the correct temperature, they'll have mostly beer with a small amount of foam on the top.
Headroom: When you walk across the room with the beer in your hand it begins to slosh around in the mug(AC signal). If it was filled level with the top, you'll surely spill it when you walk. By leaving a small amount of foam between the beer and the top of the mug, the beer can now slosh around without spilling over the top. Idealy, you would want as much beer in the glass as possible without spilling it over the top.
or.... you want as much signal (beer) into the mixer input (mug) without overdriving it(spilling beer)
