"Sucking" an egg means to make two holes in the shell and suck the raw egg out through one of them (the other one is for ventilation).
There is another phrase, "Don't teach your grandmother to suck eggs," which means "Don't try to show off what you think you know to someone who actually has more wisdom."
Here are some more contributions from our WikiReaders:
To suck eggs is like to do something really simple. For example sucking eggs. Which is not common any more but must have been when this expression was created. Today it would be like teaching your grandson how to send a SMS or Text Message. The idea is that the person, in this case the Grand mother, would already know how to do it.