Physics is full of 'imponderables'. Compared to why the speed of light is a constant despite your relative motion, and the weird principles of quantum mechanics, I rate this cosmological enigma a 5 on a scale of 1 to 10. Without a knowledge and appreciation of general relativity, you will not find an 'explanation' for this particularly compelling, but general relativity is the only 'looking glass' we have to probe the universe at large. This is what it predicts, and we are stuck with its implications until such time as general relativity is found to be wrong. A theory is not wrong simply because it 'makes no sense' to people who are not familiar with it, technically.
As for how this is possible, the mathematics are quite specific on this point. The universe had a finite beginning point for every local observer at around 10 - 20 billion years ago, and for a density less than 'critical' the universe has always been infinite in terms of the number of local observers present in space-time. At the Big Bang, the volume of a local observers space vanished, while at the same time there were still an infinite number of these observers with distinct 'Gaussian coordinates'. The classical singularity that resulted is only slightly more complicated that the singularities predicted within rotating black holes.