From personal experience I would say that the most immediate benefits of studying abroad are:
* the opportunity to learn about and experience first-hand a different culture
* the opportunity (in some, but not all cases) to learn and practice a different language with native speakers
* the opportunity to study a field in a more practical or historically/culturally relevant setting (ie, studying art in Paris, architecture in Rome, theatre in London, etc.)
* the opportunity to meet new people
* the opportunity to travel and see monuments, art, historical sites, etc.
I would say the long-term benefits of study abroad are:
* an insight into your own culture by comparison to another - it gives you perspective on why and how you do things differently from other people and reminds you that your way isn't the only way :)
* meaningful relationships with people that you either bond with through the scares and joys of being abroad - be it another student, a teacher, a host family member, etc.
As you can tell I see a lot of opportunity in studying abroad. :) One of the best things I ever did for myself was studying in France!