Why doesn't the sun explode like when you start up a BBQ grill?

Funny you'd phrase it that way. In a sense, the Sun did start up that way.

Astronomers are pretty sure in general how stars form. A large gas cloud collapses. As it collapses, it flattens out and starts to rotate. The star forms out of the center of the disk. As more stuff piles up in the center, the temperature and pressure increase (when you squeeze a gas, it heats up). At some point, the center gets hot enough and the pressure high enough that the gas (mostly hydrogen) in the core can undergo nuclear fusion.

The onset of fusion heats things up a lot. Normally, when you heat up a gas suddenly to millions of degrees, it wants to expand very quickly. That is, it wants to explode (that's what happens in your grill, though the temperature is a bit lower). In this case, though, there is a lot of gas piled up on top of the core, and that prevents the sudden expansion. The gas in the star's envelope, as it's called, heats up and expands too, but not catastrophically. It absorbs a lot of the generated heat and then radiates it away at the surface. This is like your grill too. You supply it with a steady source of gas, and it merrily burns it away, generating heat which is radiated away by the surface of the grill (well, the part that isn't absorbed by the food you're cooking).

A star is like a pressure valve in this way. The heat keeps it from collapsing, and the pressure keeps it from exploding. They maintain an equilibrium that can last for millions, billions, or even trillions of years. Eventually, the star runs out of fuel, and depending on its mass it might then explode, or just fade away.

For more info about how stars work, take a look at Nick Strobel's Astronomy Notes website. He has excellent descriptions there.