It's because of osmotic pressure, the cells try to reach equilibrium with their environment so if the difference is too big then it leaks from one side to the other (in this case out of the cells which needs the minerals). It's similar to how salt hurts snails but in the reverse direction, it seeps in when there's so much more on the outside of its membranes.

This doesn't happen just from drinking one glass of water.

Googled it before commenting to make sure I wasn't forgetting something and that's not really what happens.

Common consensus is that you're fine drinking it so long as you get the minerals you lose through sweat elsewhere in your diet

That's a myth. A very commonly repeated myth, but a myth nonetheless.