First off, infinity does exist. It is a concept. I use it all the time. I represent it by taking a number eight and laying it on its side. You can't do calculus without it. There are also many common place examples of infinity. For example, the number of counting numbers is infinite.
Second, generally string theorists describe that universe as one or more membranes (Brane is just short for membrane although "Brane" is generally used in the specialized context of a fundamental underlying structure in space-time) in which live some strings. Maybe true dreamers would like Branes to be made out of strings, but mostly they are just "put in by hand".
These strings do cool stuff. Under different conditions strings can act like different particles, say electrons or photons. In the string theory world, everything in the standard model is simply a different category of string behavior. To take an inaccurate and made up example, a string with a two hump standing wave might be a muon. The rules goving how this single kind of particle called strings act, defines all the laws of physics -- forces, particle masses, coupling constants, etc. If you knew the rules of string behavior (and true dreamers hope are simple enough to put on a t-shirt), then bam -- you suddenly can figure out everything there is to know about physics. You know all the rules of the game. This game seems likely to involve more dimensions than the three in space and one in time with which we are familiar.
Well, except that, in brane theory, you might also need to know about branes. String theory leaves open the characteristics of empty space. LQG people like to talk about spin foams. Stingy people like to talk about branes.
The main reasons that stringy people like to talk about Branes are two fold. First, Branes are a device that allows us to explain why the world seems four dimensional when physics seems to have more dimensions. If everything we know is clinging to a brain in the context of a larger dimensional universe, then the problem is solved. It also solves problems with gravity. One way to make gravity behave as it does is to allow gravitons to slip off the brane, thus we know why it is so weak. It isn't really weak, it just mostly doesn't arrive at a destination on our brane.
I'll close by noting a key point. No body has made an useful testable predictions using string or brane theory. It is untried by experiment. We aren't even sure that it actually reproduces obvserved physics although we have some pretty suggestive lines of reasoning to think that it does or comes close. The brane part of the story is even more speculative than the string part.
