./configure means: run the shell script (or binary) named
configure in the current directory. You'll execute whatever is in that script, which could be: well, anything you can imagine. Usually
configure scripts are auto-generated from trusted inputs, but if you're downloading something random from the Internet, there's absolutely no guarantee that
configure can be trusted.
make also runs a script, probably
makefile in the current directory which is written in the make language. It can also do pretty much anything.
Practically: if you compile a program once as root, then as you fix compile problems you will need to keep compiling as root, because a normal user won't be able to clean up all the intermediate files that are generated. So it's not uncommon to see someone recommending you build as root because one time it worked for them when building as user didn't, generally because they hit this sort of situation and didn't realize it.
If you don't trust the source (and I imagine you are, since you're posting on Security instead of Unix Stack Exchange), you can compile (and install if you want) and run within a chroot, container or VM, so you never need to directly give the software root on your main system (though programs can theoretically break out of any of those jails).