Why does General Relativity with no dark energy, no cosmological constant, predict that the universe should expand? All the matter in the the universe causes collapse of spacetime, not expansion. So, where does the "expansion" come from? An intuitive answer is appreciated.
It isn't strictly true to say that GR predicts the universe should expand. What GR predicts is that a static universe cannot be stable.
This is a lot simpler than you might think. Suppose you see a stone in mid air. Experience tells us that stones don't hover in mid air, so someone must have thrown the stone and it must be on its way up or on its way down. If we measure the velocity of the stone we can tell which of these is the case.
Likewise if you see space full of galaxies this can't be a stable configuration because if the galaxies are all stationary they would attract each other. If the galaxies are there it's because some thing threw them (we'll get back to what I mean by threw in this context). By measuring the velocities of the galaxies we can tell if they are on the way up or on the way down, and if we measure the velocities we find they are on the way up i.e. the galaxies are all receding from each other.
What GR does is make this quantititative i.e. with a few simplifying assumptions we can calculate how the density of matter behaves with time. And when we do this we find in order to get where it is right now all the matter must have started out infintely dense and moving away from all the other matter infinitely fast, and that happened 13.7 billion years ago. Obviously, this moment is the Big Bang.
The key point is that GR isn't the reason the universe is expanding. It's an experimental observation that the universe is expanding, and starting with this observation GR gives us the means to calculate how the universe behaved in the past and how it will behave in the future.
The reason the universe is expanding is because of whatever happened at the Big Bang but GR can't explain that. Hopefully some theory of quantum gravity will explain why the Big Bang happened, but until then we just haveto accept that it did and use GR to calculate what happened next.