First of all, not all of them are round. Mostly they are. And this brings up the main reason most of them are round, in a way.
A round manhole cover cannot fall through the hole. Even if you lift it (which is difficult without help because the covers can weigh up to 200 pounds) and turn it up vertically and drop it into the hole, it won't fall through. The reason why is that a round cover has the same width all the way around; and, the same diameter. A square cover would not have the same width all the way around. The diagonal measurement from corner to corner is different than the sides. If you picked up the cover, turned it vertical and then diagonal, you could drop it right through the hole.
This is not to say there aren't other shapes that might work. The Reuleaux Triangle, for instance. This shape is a triangle that has sides that come out a bit and are rounded. Also, it's possible a square cover could have a bit of material coming out a couple inches from the sides to prevent the cover from being dropped into the hole. However, seems much easier to just make a round cover.
For one thing, a round cover is easier to put in place; this is another possible reason why most of them are round. You don't have to adjust for corners and angles; just plop that thing in place. Pretty efficient. I'm sure city workers' time is valuable, like it is for the rest of us.
It is also easier to move a round cover around; just tip it on the side and roll it.
It is also said that it's easier and more cost-efficient to manufacture round covers. This makes sense. You don't need to use as much material, and you just have to make one basic, round piece.
Another consideration is the fact that a manhole cover covers what is essentially a pipe that takes the streets' run-off away. Pipes are generally round and so their covers are round. Pipes are round, also, because being round makes them more efficient and cost-effective. Round pipes more efficiently carry material. Again, it requires less material to make round pipes too. They have a stronger structure and handle stress better, also.
So, there you have it, a few soundly reasoned answers for why manhole covers are round.