Social
In the course of learning ballroom dance, you'll meet fun active people in your classes and venues where you practice. You will have lots of opportunities to make new friends.
You'll gain confidence in social situations. You won't feel left out at weddings and parties where dancing is part of the fun. You'll have the confidence to ask someone to dance, or to accept when someone asks you!
Relationship with your Partner
The most important benefit of ballroom dancing for me is the effect on relationships.
If your partner is your spouse or longtime friend, you'll find that dancing adds another dimension to your relationship that is easy to overlook in the day-to-day details.
Dancing causes you to set mutual goals and dedicate time each week to accomplish them. The romance of getting dressed up and going dancing together (even for a class) reminds us of why we're together in the first place, and it's not just because of the kids and the bills! It brings a whole new level of respect for your partner and makes you see them in a different light.
If you've had disagreements, it forces you to set them aside for an hour, and generally that hour will put the trivial back in perspective.