I understand the evidence for the big bang (expansion, background radiation, etc), but how do we know that it was the start of the universe? Why couldn't it have occurred in an existing, but very sparsely populated universe?
We don't know what's on the other side of the Big Bang, if anything at all. The Big Bang theory as well as the rest of physics is agnostic on this possibility. During the time period after the Big Bang, the entire universe was basically in a hot, dense plasma state; any physical signals propagating from a period before the Big Bang that could potentially influence the present or provide us with information of a pre-Bang past would be eradicated by the extreme conditions.
So the singularity of the Big Bang is effectively the start of our universe because it is effectively an impenetrable cosmic firewall disconnecting us from anything that came before. Maybe there's another room on the other side of the wall, or maybe it's just more wall. Until someone comes up with a more refined theory or an experiment finds a way to detect a tiny signal that somehow made it through, it's pointless to speculate.
We don't is the simple answer.
However it is difficult to test any such theory and so it veers in the direction of metaphysics.
But one theory is that the physics in our neighbourhood changed (a state change, much like a change in the quantum state of a molecule migh lead to a release of energy as light) and that the energy released caused the early rapid expansion of the universe.
The current model of cosmology starting with the Big Bang states that all spatial and temporal dimensions (Length, width, height, and time) as well as the four fundamental forces originated from a single point. With time starting with the big bang, there was no 'before', just like there was no up or down, no gravity or electromagnetism. There was no existing universe when the big bang occurred. It happened, and then the universe existed (causality).
So to say that it started in another universe would require a different hypothesis than the current cosmological model.
There are many challenges to formulating a hypothesis that could explain what you are asking. Some of these challenges are how to account for the laws of conservation of mass and energy. For example - if there was a 'sparsely' populated pre-existing universe, where did the mass and energy for our universe come from? Was it always there? If so it must have been in the form of some type of black hole. This much should be self evident - we can have black holes that exist in our universe with only the tiniest fraction of the total mass. If ALL of the mass were concentrated in one place, then it too would exist within it's own Schwarzschild radius. As a side note, it's a common exercise in college astronomy classes to calculate what the observable universe's Schwarzschild radius is. Surprisingly enough, our observable universe can be computed as existing within an event horizon, but these calculations require that there be a lot of assumptions of total mass and maximum extent of the observable universe (which usually limits the diameter to being smaller than what the Schwarzschild radius is prior to starting the experiment).
Another difficulty might be trying explain a black hole suddenly going into reverse and ejecting all matter/energy, spatial dimensions, fundamental forces, and time into an existing universe. How does that explanation then exclude all of the black holes that are thought to exist in the present epoch of our universe?
In short: We don't know for a fact that the universe we see today didn't already start in a universe which was in an existing state. However, to the best of my knowledge, there have been no attempts made to explain how a sparsely populated universe could suddenly grow a densely populated mini-universe inside of it. Any attempts that are out there may not have gained wide enough audience yet to include me.
Also, remember that for a hypothesis to become theory, it must be testable. There is no way to test events that occurred before the big bang. Therefore there is no possible way to test for the sparsely populated universe you asked about in your question.