I am just wondering because I don't want to do anything to my browsers that would be bad.
Should be safe! It's never caused me any problems with Firefox (or browsers based on it), and no problems with Chrome/Chromium based browsers.
The cleaning it does in Firefox was, and may still be a command you could run yourself inside of Firefox, it's just far more convenient to have a cleaning program automatically do it without messing with a long line of code. Thing is unlike SpeedyFox it won't compact the databases ("Sqlite") files every time you run CCleaner, it does it on its own whim and time.
I seem to recall that there was a problem with the way CCleaner worked with the Firefox sqlite database, - about a year ago?
Shouldn't be an issue now.
I never had an issue with it and Firefox because it would only rarely compact the Sqlite database.