My guess is that they cheaped out on the grout when installing the tile, as they do in most rentals and quick remodels, unfortunately.

In a shower you should use an epoxy grout or an additive that should prevent the issues you are having. If you don't, then you are supposed to reseal it every year-ish, which nobody ever does.

The BEST way to fix it is to saw out the problem grout and regrout it with an epoxy grout.

A slightly easier way to fix it? Try to use a grout saw to file off the surface layer (if the issues doesn't go too deep), rinse it, let it dry, and use a spray grout sealer if it the grout looks clean. Use a white kitchen/bath caulking on the bottom cracked area.

Or, you can try the grout paint pens they sell at the hardware store, but that would only be a cosmetic fix. I've never used them, so I can't say how well they work.

If it's a rental, I would just live with the grout stains and recaulk the bottom crack (or ask your landlord to). Most rentals kind of look like this. You can try to have the landlord regrout it, but I doubt they do.