General Relativity is a theory that explains the effects of gravity (i.e. the mass of all things upon the motion of all things) in a geometric way. The geometric nature of the theory leads to the term 'curvature'. As Amanheis suggested, the only way to know if this is the correct theory is to do experiments, and experiments have so far agreed with GR, and hence space-time must curve in the way implied by this theory. That is as much as we can say. It is possible that we could re-formulate GR into a non-geometric theory that was mathematically equivalent and then we wouldn't have 'curvature' of space-time. The notion of curvature is a mathematical construct that gives us the correct answer.
By the sounds of you original post, you've probably been told something like this before but it didn't really satisfy you. Unfortunately your request for an answer 'proved not by maths or theory - but physically and factually' makes no sense. What does it mean for space-time to curve 'physically'? The physical theory, that cannot be separated from the language it is written in (mathematics), describes the effects of gravity as space-time curvature. The theory gives the correct predictions when compared to experiments, and therefore the theory appears correct (at least in the regimes it has been measured in). There is no other way to test a theory more 'factually' than this I'm afraid. Rememeber that space-time is a mathematical construct, so therefore anything is does (such as curve) is also a mathematical construct.
Note that Chronos has misunderstood your question, he is referring to the question of whether when we use a particular solution (the Friedmann-Robertson-Walker model) of GR to describe the large scale Universe, is the spatial part flat or curved. This is a different question, and in either case space-time is curved (even if the spatial part in a particular co-ordinate system is not). If this is confusing then don't worry, but you should realize that the jury is in no way 'out' on the question of space-time curvature explaining gravity, i.e. regarding the bending of light around massive objects. As I say, Chronos is answering a different question, so don't let that confuse you.
On your FLT ideas, note that just because space-time can be curved in GR is not the ticket to a free lunch. Curvature is caused by the presence of mass-energy, so you don't make curvature out of nothing, and you cannot curve space-time in arbitrary ways. The only know sensible description of a warp drive type situation in GR is the http://en.wikipedia.org/wiki/Alcubierre_drive" however it is important to note that this 'solution' to GR requires an energy source that violates many many rules of nature (it have to have a negative energy for starters, which is a non-sensical concept physically). If I was going to write a Sci-Fi story using FLT it would use an Alcubierre drive, but it is still literally impossible based on the current laws of physics. So, as I say, space-time curvature does not mean 'anything goes', the universe still behaves lawfully.
