A bridge is exactly what it says. You use it to connect one part of your song to another. Most songs use it after the second chorus. Instead of going back to a verse, they use a bridge, and then go back to the chorus again. (A= verse, B= chorus, C= bridge). Sort of like this.

A B A B C B

There really is no theory behind writting a bridge. All you really have to think about is writting a different part, that connects two parts that don't go together without putting something in.

If you look at the form above, the bridge is used to connect B and B (of course without a bridge it would just be B repeated x amount of times). Looking again A is (the verse) is used to connect B and B before. SO you could rewrite the verse, and get a bridge.

A few ways you could to that.

Change the harmony by using subs.
Keep the same chords but change the rhythm and overall style.
Progress from the last chord in the chorus to a new key and continue.
...and the list goes on.

Also, a bridge doesn't have to mean writting a completely new part. Take harmony for example and say you wanted to go from V to iii. You may not like that (I don't), so you could add another short part (1 or 2 measures) to connect them, just by doing this.

| V . . . | iii . . . . |
to this...
| V | vi . ii . | iii |

So when thinking of writting a bridge, remember your aim is to write enough music to make to parts that don't go together go together. Use your own creativity in doing so.