Eggs can be found randomly throughout the levels as you play. In order to crack one you need the corresponding colored Hammer. so if you picked up a white egg you need a white hammer to crack it.
You can either purchase the hammer in the store or you can sometimes find hammers in the levels. Once you have a match you have to go to the menu where you see the Eggs and the words Match below them. Go there and crack the egg until it opens. You get a prize and the achievement pops.
FYI you can only have one egg of each color at one time so it's in your best interest to crack them ASAP in case you come across another of similar color.