To hatch the eggs, match blocks in the spaces next to them. They can also be hit with boosters. Eggs require two hits - one to crack the shell, and another to free the bird inside.
Sometimes, a batch of eggs will appear inside a nest. Matching on any adjacent block, or hitting any part of the nest with a blast will cause one egg at a time to crack or hatch. Once you’ve hatched all three eggs, the nest will disappear from the board.