AppleScript to Rename File Names
The Applescript-native way to rename files is:
tell application &quot;Finder&quot;
    set theFile to (choose file)
    set name of theFile to &quot;Yay&quot;
end tell

If you wanted to rename every file in a folder, I'd put the files into a list and use a Repeat block on each one:
tell application &quot;Finder&quot;
    set theFolder to (choose folder)
    set folderContents to every item of theFolder
    repeat with theFile in folderContents
        set name of theFile to &quot;Yay&quot;
    end repeat
end tell

(This code won't actually run as-is because you can't have two files named &quot;Yay&quot;, but the first does get renamed, and you get the idea.)
This is closer to what you want to do, but we're not quite there—you want to get the current file's name, make some changes to it, and save the new name.
There's a subroutine I found years ago somewhere (I can't seem to find the source now) which will get everything to the right of a certain character, such as an underscore. This makes it easy to do exactly what you want. The full, final code is:
tell application &quot;Finder&quot;
    set theFolder to (choose folder)
    set folderContents to every item of theFolder
    repeat with theFile in folderContents
        set oldName to name of theFile
        set newName to my rightString(oldName, &quot;_&quot;)
        set name of theFile to newName
    end repeat
end tell

on rightString(str, del)
    local str, del, oldTIDs
    set oldTIDs to AppleScript's text item delimiters
    try
        set str to str as string
        if str does not contain del then return str
        set AppleScript's text item delimiters to del
        set str to str's text items 2 thru -1 as string
        set AppleScript's text item delimiters to oldTIDs
        return str
    on error eMsg number eNum
        set AppleScript's text item delimiters to oldTIDs
        error &quot;Can't rightString: &quot; &amp; eMsg number eNum
    end try
end rightString