How do you remove the honey from the honeycomb?

There are various ways. The most usual way is to slice off the wax capping from the cells, place the frames (the comb is in a wooden frame) into a device not too unlike a spin drier and the frames are spun. The honey is thrown out of the cells by centrifugal force on to the sides of the container and drains down to the bottom. This method has the advantage that the comb is unharmed and can be returned to the bees for re-use.
If you don't have a honey spinner, you could simply crush the comb to release the honey and sieve it to remove the wax, or you could gently melt the wax from the honey using no more heat than absolutely necessary and they will naturally separate. The disadvantage of these methods for a beekeeper is that the comb is destroyed in the process.

Another way is to uncap the cells and suspend the frame, upside down, over a bowl and wait for the honey to drain out. This is a slow process, however. The frame has to be upside down because the bees don't make cells horizontal in the comb as they build it, but slope them upwards from the centre, like a shallow 'V'.