With a multi-bladed razor, the leading blade lifts the hair slightly as it cuts it. Before the hair has a chance to retract, the next blade reaches it and cuts a little lower down. This results in a shave that is not just flush with the level of your skin, but actually slightly below it.
Naturally this is subject to the law of diminishing returns, but a razor with more blades also distributes the pressure of the blade cartridge more evenly. There is some evidence that this keeps your skin flatter so that you get a more even shave and less chance of nicks and cuts.
So it is worth going for a multi-bladed razor, but whether you’re better off with three, five, or even seven blades is going to come down to what feels most comfortable for you. Almost all scientific research in shaving technology is sponsored by one of the razor manufacturers, so a truly objective figure for the optimum number of blades is hard to come by.