Einstein's Spacetime has four dimensions. If the size of one of these dimensions is zero, then the four-dimensional 'volume' - or whatever the corollary to 3D volume is called in 4D - would be zero.
Is that enough to explain the observations that lead to the Big Bang theory, rather than an infinitesimal 'point'?
It is a common misconception in science popularization that "the universe was contracted to a point at the big bang". It is ok to think about it this way to get a picture, but it's not really what the math tells us was happening in the beginning of the universe.
Without going into too many details, the universe in Einstein's theory of general relativity is described by two things: a manifold and a metric tensor.
The manifold
You can think of the manifold as every point in spacetime, it looks like a 4-dimensional sheet that can bend, stretch, etc. For example, if we actually lived in a 2dimensional space + 1 dimension of time, the manifold would just be an infinitely big cube! each slice of the cube would be all of space, at different times.
The metric tensor
The metric tensor is a grid over that manifold, if you give me any two points in the manifold, I can use the metric tensor to figure out what's the physical distance between the two points. It's basically a ruler for each point in space.
For example, if Alice is standing at some point in space, and she sees Bob at some other point in space, she can use the metric tensor to calculate the distance between them and conclude that they are, say, 5 meters apart. If they don't move and she repeats the calculation at some later time, she might find that they are actually 6 meters apart. The notion of an expanding universe has to do with this "grid" expanding with time, not with the manifold itself expanding with time, since neither Alice nor Bob actually moved from their respective points in spacetime, it's just that the physical distance between them got bigger.
So what's the big bang then?
The big bang is this grid collapsing to zero. In other words, it is a point in time when the grid breaks down and tells you that the distance between any two points is zero. That doesn't mean that the universe is a point, since I can still speak of different points in the manifold, it's just that their distance goes to zero.
An immediate consequence of this is that the big bang actually happened at every point in space, not in just one tiny point.
Quantum disclaimer
Have in mind that the fact that our best physical theories tell us that the grid broke down and that every point was infinitely close to every other point means that our theories are probably wrong! At very small distances we should be using quantum mechanics, and the theory of general relativity is not compatible with quantum physics, so we need a full quantum description of general relativity to know what really happened at the big bang!
I hope this helps!
