A vector in that space represents the state of a physical system. It's time evolution is given by the Schrödinger equation.
In case you need something to compare that to: In classical mechanics the state is given by a specification of the location and velocity of all component parts, and the time evolution is given by Newton's second law.
If you assume that the state vector is an accurate description of reality instead of just a mathematical tool (needed to calculate the probabilities of the possible results of an experiment), what you get is the many-worlds interpretation of quantum mechanics. So is that assumption/interpretation correct or not? The question is actually irrelevant. It's kind of like asking if these three dots (...) are really two dots to the left of one dot, or one dot to the left of two dots.
