When you normally boil water, you increase the temperature of the water, which increases the water property "vapor pressure" until it becomes equal to atmospheric pressure at 100 C. Or if you are at Mt. Everest, where the pressure is lower, water boils at 71 C.

When cavitation occurs, the opposite is done: you lower pressure until you reach the vapor pressure of the liquid at the current temperature.

In supercavitation, you create a shockwave in front of the bullet/torpedo/propeller which raises and then lowers pressure such that the water boils/cavitates. Why is this good?

Because it lets you put most of the velocity difference between the bullet and the water inside a gas layer. This gas layer has much less resistance to velocity differences than water. Running in water is much harder than running in air, right? So far so good.

So what happens when a supercavitating bullet is fired at greater depth? The pressure is higher, so it's harder to create cavitation, because you have to lower the pressure more to get to the vapor pressure.

Why is that good? Doesn't it reduce the nice gas layer? Yes, it does. My guess is that it also reduces the length of the cavitation bubble behind the bullet, and that reduction in drag is bigger than the increase in drag from a thinner gas layer.

In the older guns, the trailing gas bubble is much shorter because of the projectile shape, and the reduced bubble length at greater depth causes part of the projectile to be outside the bubble, increasing resistance.

You could probably put some decent numerical estimates up by spending an afternoon with White's Viscous Fluid Flow and a steam table.