A couple of issues you may have.

Firstly for your immediate issue it says that it's missing something. If you look closely in the error messages you'll notice this:

Configuration failed because mysql-connector-c was not found

And right after that you'll see:

Try installing: ... <various packages>-dev

The problem is that installing R-packages on Linux often involves compiling from source to a greater or lesser extend. In this case it needs to compile the RMySQL package from source. In order to do that it needs various MySQL sources which are not provided with R or the R-package but must be installed independently. A source package is almost always named <name>-dev or <name>-devel. In this case the error is quite specific and helpful. In other cases it may just present a list of requirements where at some point there will be a "no" and you have to go hunting for what specific package it's actually complaining about.

Also versions are important, so you may think you have it but it's the wrong version - this can be tricky as having several version may lead to conflicts.That's an issue if you want to stay up-to-date with R but you system isn't for some packages.

For MySQL that shouldn't be an issue though IMO.

The other issue you may have is that you as it appears are trying to install in the a root owned dir where you don't have write permissions as a normal user. To overcome this you either start R as root or you make a local R dir in your home for your own installed packages. If you start R as root please be careful not to save your session when you close because you now have some root owned files that R will try and fail to write to when you start as normal user, in particular ~/.RData and maybe ~/.RProfile. You can just chown them back to yourself, it's just another gotcha worth to know about.