It means that he dreamt so much that the fantasies he dreamt of could no longer be held in- they were becoming realities.
Explanation:
To burst at the seams means is an idiomatic expression that means, to overflow. When the song composer of <em>Fireflies,</em> Adam Young stated that his dreams were bursting at the seams, he meant that there was a lot of things he dreamt of. These dreams were overflowing or becoming so much that they could not remain as just dreams. They were now translating into realities.
This can be further seen in the lyrics that 'he saved a few and kept them in a jar'. This means that some of his dreams that he wished to become true were etched into his memory.
