Originally, the easter bunny is a hare, not a rabbit.
Rabbits give birth inside burrows and their young are blind and helpless. Hares give birth in nests in grasslands and their young are ready for live immediately.
Birds lay eggs in nests also in grasslands and after coming out of the egg, the leftover shells are there. So for a long time the mythology was that hares came from eggs also.
On how the christian mythology came interlinked with this, that is explained in other comments.
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