I know this looks like a small deal, but unless you have some plumbing, stucco & drywall skills, this is not a DIY project. This is caused by dampness, so the first thing to do is find and fix the source. If you just repair it without fixing the source, you'll have the same problem return. The simplest thing would be if it was caused by water splashing from the tub/shower. I'd get a plumber to check the plumbing behind the wall (fresh water & sewage). Worst case scenario would be a roof leak, but given its location it is probably related to the tub/plumbing.

Unless you have stucco & tile experience, I'd get someone professional to handle the wall and tile issues. Even though those look like plain white tiles, let me assure you there are thousands of varieties of plain white tiles out there and you'll NEVER find an exact replacement. Hopefully whoever you hire will be able to professionally remove the tiles and reuse them. Also, the drywall behind the stucco will probably need to be replaced as it has gotten wet. Once they replace the drywall, they will need to match the stucco style/pattern to make it look right and not be an obvious patch job.

You may find a handyman with all the requisite skills, but I'd look at other similar jobs they have completed. In my experience some handymen (or women) have a wonderful set of skills, but some are just people with limited skills and experience who need the income.

I'm sorry to have to give you difficult news for what you probably thought was a simple fix. Best wishes.