my reasoning is that it is not possible to achieve the infinite, but it is quite a handy point of reference. That is to say, that if we can prove (mathematically) that something holds for any number less than infinity, then it holds for every number. if an equation has no "limit" (as in, as the sequence continues, the value increases, either at a constant or non-decreasing rate), it is easiest to say that it has its limit "at infinity". that isn't to say that it ever reaches infinity, but rather that it will keep getting bigger indefinitely the further the sequence goes.
Similarly to say that Space is infinite is to say that no matter how far you were to travel, you would never reach an "edge", because it is expanding at a rate greater than we can travel. It would be easy, at that point, to say that space must then be finite, if we can potentially travel to the edge of the universe (which by logical reasoning we must, since the universe was definitely once of finite size, according to the big bang theory), but that then requires that there be space for the universe to expand into. But the universe contains everything in existence, so how can it expand if there is nothing to expand into? Well, consider the arithmetic of infinity. Infinity + anything is still infinity. Similarly infinity - anything is still infinity. So that means if the universe is infinite in size this handy conjecture regarding adding and subtracting anything from infinity helps solve these problems :P
Of course, we could just assume that rather than the universe getting bigger, we're getting smaller, thus giving the impression of expansion.