What he means would be easier to show by sketching the animal on a piece of paper than describing it here, but I'll try.
Imagine that you connect the food's entry point (mouth) to its exit point (anus). Now, for there to be a digestive system the path along which the food would travel would have to take up some space, right? All parts of the digestive system, which is what Hawking refers to with gut, would have to have a certain width.
This path would divide the two-dimensional animal like a canal where water can flow inland from one point at the shore of an island to another point at the opposite shore (or somewhere else, doesn't have to be opposite, really).
The moment you cut this canal, you make this digestive path, all across from one end to the other you have effectively cut the animal in half and you could pull it apart.
Does that make sense?
ETA: I found a picture that may help to illustrate it. It's a rather crude sketch but that fact might actually be helpful in this case.
It's a rough sketch of a cow's digestive system. Now, there are some loops in the various stomachs of that cow which would not be possible in a two-dimensional animal.
If we ignore the loops in the middle you could effectively pull off the upper part of the cow from its lower part because of the digestive system.
That becomes most clear when you look at the cow's head. The mouth and throat separates the upper jaw from its lower jaw. You can imagine this to go on right through the cow to its anus and you can pull the cow apart.