What do two flats mean at the start of music?

The two flats at the beginning of a piece of music indicate the key signature. The key signature tells the performer what notes to play depending on the notes they use, to ensure the music stays in tune.

In this case, the two flats indicate the key of B-flat major or G minor. Every B-natural needs to be played as B-flat, and every F-natural needs to be played as F-flat.