It's not about traveling. It's about staying as long as he wants to and whenever he wants to. American citizens are limited to staying 90 days (in any 180 day period) in the EU as tourists. An EU passport would allow him for example to take a 6-month vacation in France.
EDIT: The 90/180 rule is for the Schengen area, not the whole EU, and, yes, that includes Switzerland, which is not in the EU. Not sure what the rules are for the rest of the EU, but he probably cares about Schengen anyway.
EDIT: The 90/180 rule is for the Schengen area, not the whole EU, and, yes, that includes Switzerland, which is not in the EU. Not sure what the rules are for the rest of the EU, but he probably cares about Schengen anyway.