It's called water glassing eggs. It can be done with pickling lime (calcium hydroxide). Some people use sodium silicate, as well, with some people saying they only need to paint sodium silicate onto the egg.
The eggs have to have the bloom intact - they can't be washed (supposedly, I am curious to test this in the future). The minerals plug the pores on the egg and prevent oxygen exchange as well as moisture exchange and any bacterial contamination (basically, the shell is reinforced to take over for the bloom).
The eggs will not be perfectly fresh - many claim they are runny/watery after being in the solution for a long period of time, but work okay for omlettes and recipes requiring eggs.