Miles2Go2 said: "Wow. This is hard to put into words, but I'll try. Obviously, I can only speak for myself.
Theater (whether it be of the musical or straight play variety) is an art form. As such, at its best, theater is an interactive process between artist(s) and audience that attempts to reflect truths of the audiences' lives back to them in a way that resonates. Of course, some musicals/plays accomplish this better than others. And because we all attend musicals/plays with our own backstories, all shows are bound to resonate more with some over others. You see this every day on the message boards where some posters passionately post about their favorite show(s) while others have an equally passionate response in the other direction to the same shows.
As others have alluded to, there is something especially cathartic to viewing live theater in a room with hundreds of other people who are there for the same reason and experiencing those moments simultaneously with you. Of the art forms, live theater is special. Listening to recorded music, watching tv, seeing a movie at the cineplex, reading a book, or viewing a painting or sculpture are all capable of providing unique cathartic moments, but those art forms are static in they don't change although our reaction to them may change with years or increased exposure. The only art form I can think of other than live theater that reaches this intense interactive cathartic experience is a live music concert, but even those are two different animals. Rock concerts encourage us to be as outwardly interactive (within limits) as we want with the musicians while live theater encourages us to mostly be interactive in a internal way (other than spontaneous laughter and tears). In that way it requires us to be actively and internally engaged emotionally and intellectually with the artists. As a result of not outwardly showing (as much) my reaction, it forces me at least to interact at a deeper level, which allows the best art to pierce me deeply. In that way it includes the best elements of all the other art forms IMHO. I suppose you could say that performance art and immersive theater experiences could also be included here, but so far they don't mirror the experience I have sitting in a theater watching a live musical or straight play performance.
I'm not religious, but I suppose people attend live theater for some of the same reasons people attend places of worship. Namely, to seek a deeper way of understanding their lives and, more importantly, the lives of the people they share this planet with.
Sorry, this was so wordy and I hope it makes sense to someone besides me. 
I just realized you asked specifically about Broadway theater. I would say all the above applies to Broadway, but in addition when I consider all the obstacles that have to be overcome, all the forces that have to align and how many people will never make it to that theater in New York City to see that specific play, I feel very honored and lucky to attend a Broadway show. It heightens the experience even more.