It is a great workout
Ballroom dancing is not for the faint of heart. It requires a dash of determination and a splash of stamina. If you lack the latter, do not worry. You will gain some.
Many of the dances in ballroom are cardio workouts with good form and figures put to music. Quick paced dances, like quickstep, mambo, jive, cha cha, and samba, burn through calories (and, resultantly, fat), especially the more often you do them. They train your body to more efficiently draw in oxygen and circulate it through the body, thus resulting in a greater lung capacity and stronger heart.
Even in some of the slower dances, there is a great deal of muscle training as you have to properly activate some of your (sometimes little used) muscles to hold certain shapes or execute particular moves. If anything, holding up your frame is a shoulder workout in and of itself.
It helps form or strengthen social bonds
Since ballroom dancing is a social activity, it requires socializing. Whether it be with a friend you dragged along with you or a stranger you just met when you showed up, you will find yourself interacting with various individuals in a (hopefully) low pressure environment.
You cannot help but crack a few jokes or share a few smiles when you are all trying to fumble your way through new movements, and long before you know it, you will find yourself with a few new friends to tango out in public with when just the right song plays.
It boosts memory
Ballroom dancing requires great mental and muscle memory. Each time you dance, as you learn new styles, various figures, proper technique, and (potentially) whole routines, you are constantly challenging your mind and body to keep up and get with the program. You will find yourself recalling easily that the hand-to-hand comes after the spot turn right after three New Yorkers. What is even more interesting is when your mind blanks out completely, but you find yourself executing an underarm spin when you feel your partner's hand twitch a certain way long before your mind catches on.
It improves interpersonal skills
Your people skills are forced to improve even if it is just for swing because ballroom dancing is a partnered activity, and you have got to work with others. When you are dancing, you and your partner(s) have to learn how to communicate both verbally and nonverbally if you want to accomplish anything. It is all about coordination and learning to trust, compensate for, and improvise with each other. You will find yourself knowing exactly what to do with the subtlest indication from your partner even if it is not part of the routine.
It builds confidence
Because of the physical benefits of ballroom and the capabilities you are gaining as you continue to learn and practice, ballroom dancing does begin to improve your confidence and self-esteem. Be it that you are getting more in shape, understanding your body more (and knowing how to move it), and/or gaining an interesting new ability that you can pop out when needed, you might find yourself holding your head a little higher, even if it is only when you are slipping on your dance shoes.
It is liberating
Such an environment provides an arena where you are allowed to mess up and learn. It is okay. You are not meant to get it right off the bat (something I have to tell myself every day). You can make mistakes. You can trip, step on a few toes, or spin too late, but do you know what is so great about that? You are not alone, and you will get better.
It is fun
You have an excuse to move your body to music and (maybe) look good while doing it. You can call it whatever: exercise, training, etc., but whatever it is to you, you should be enjoying it as you do it.
It prepares you for the most improbable moments
You know... that moment when that particular song comes on and you make eye contact with a stranger across the dance floor. You both step towards each other and fall right into frame before gliding away to the shock and awe of everyone around you...
...or, you know, when someone grabs you as you step out of the staircase and starts foxtrotting down the hallway. True story. It was great.
So... what are you waiting for?
