Android - Does rooting erase anything?

will that erase anything on the phone

Rooting itself should not erase anything (except for, maybe, temporary files created during the process). However, on some devices rooting requires to first unlock the bootloader -- and that usually includes a factory-reset (which means: all data and user-installed apps are gone -- which you can work around on Android 4.0+ performing an adb backup –apk –shared –all –f /path/to/backup.ab before performing the unlock, and adb restore /path/to/backup.ab afterwards, as Compro01 pointed out in below comment).

Will it change anything other than the ability to run app(s) as rooted?

Yes.¹ As the necessary binaries (usually su, SuperUser.apk/SuperSU.apk, and busybox with all its symlinks) are pushed to the /system partition (which otherwise is mounted read-only), it will change that partition. Might sound trivial now, but the thing is: some ota-updates (and other updates shipping directly from the manufacturer/provider) first compare the partitions checksum. If that doesn't match (and it won't after rooting), updates might be refused/failing.

On the other hand: if those updates wouldn't fail, you might lose root when they are applied. If they include a full /system image, that is.

---

¹ _{Update: With Magisk the changes described here are not applied to the system partition.}