Broadly speaking, the fitter you become, does your maximum heart rate increase, decrease, or stay the same?

Your max HR stays the same regardless of fitness level. It depends only on your age, and it decreases with age. You can calculate it without going to max effort by doing 220- your age (keep in mind this is plus or minus like 11).

That being said, your heart rate you work out at will be lower for any given intensity and you will be able to work out at a higher % of your HR max without getting fatigued.

"Maximum heart rate" is a strange concept.

When you are fit, your heart rate tends to not be so high when you exercise, but your normal pulse would tend to be the same or possibly lower. As to what your maximum rate is, who knows?

I assume it must mean the rate at which your heart beats before it bursts. Otherwise the expression is meaningless, or at best a misnomer.

Generally as you get more fit, your body works less. So your hearbeat at rest should get lower.