To get a broken light bulb socket out of the fixture you first turn off the power!!! VERY IMPORTANT!!!!
Carefully break away the glass part away from the socket. Use gloves and safety glasses in this instance because the bulb is hanging down.
Using needle nose pliers, "gently" try to wiggle the center remnants of the bulb while attempting to turn the bulb counter clockwise. It's probably going to break off, but it's worth a shot.
If the center remnants break off, then you will be left with the socket. Open the needle nose pliers are far as they will go so that the back of the needles compress against the inside of the socket. Now try to turn the pliers counter clockwise while keeping pressure on the socket with the pliers. Hopefully that will get it out.
If that doesn't work, then you would need to grab the lip of the socket with the pliers and try to turn it that way. That often damages the socket and you may have to cut it out in pieces using tin snips.
The reason these sockets get stuck is usually moisture. That causes the socket to corrode and makes it difficult to remove. If you have a wet area (like a bathroom or even an outdoor fixture), it is a good idea to coat the sides of the socket, on the new bulb, with a light coat of petroleum jelly. This will usually prevent this from happening again.