I assume you have the bash shell on your mac.
You can do it using the bash parameter expansion:
for name in *; do mv -v "$name" "${name#[0-9]*_}"; done
This will remove all digits up to the first _.
Note: this will overwrite files which end up having the same name.
Example:
$ ls -1
000_file0.txt
001_file1.txt
002_file1.txt
003_003_file3.txt
$ for name in ; do mv -v "$name" "${name#[0-9]_}"; done
000_file0.txt' ->file0.txt'
001_file1.txt' ->file1.txt'
002_file1.txt' ->file1.txt'
003_003_file3.txt' ->003_file3.txt'
both 001_file1.txt and 002_file1.txt will be renamed to file1.txt. in this case the file 001_file1.txt is overwriten by file 002_file1.txt. which file will get overwriten depends on the ordering of the * glob. typically it is alphanumerical.
if the separating character is some other character then replace this character
for name in *; do mv -v "$name" "${name#[0-9]*_}"; done
^ this character
for example space
for name in *; do mv -v "$name" "${name#[0-9]* }"; done
^ space