As a society, we in India tend to LOVE drama & hype. This love manifests itself in almost every aspect of what I think is a fairly loud culture.
The big bold flavors in our food, the use of gold & silver vark to decorate sweets, Ekta Kapoor's daily soaps, the Kumbh Mela, IIT, Tirupati, our weddings, the IPL all are a sort of personification of this ingrained love.
Since bollywood is the prime form of fiction for a large section of the country, it is bound to reflect this love. The superstar roped in to play the lead, crazy "action" sequences, enormous number of background dancers, blaring loud costumes, unrealistic over the top plots, villains like Magambo & Gabbar and obviously the songs.
As has pointed out, music is a part of every step of life in India & hence songs have always been a preferred way to accomplish almost any task. From highlight points in narratives, dreaming, growing up & emotions to adding some fun, give the audience a break & increasing revenue, a song can do it all.
Not so surprisingly, the moment we were given audio, songs in movies seemed like a very obvious thing to do. Alam Ara was India's first movie with audio. In the words of Shyam Benegal,