“We also say that space and time both started at the Big Bang and therefore there was nothing before it.”
The statement that space and time did not exist prior to the Big Bang appears to present a logical impossibility, at least with respect to time. Any physical event implies change, and change presupposes — or would logically require — the existence of time. Or not?
The question of what happened before the big bang is a long-standing mystery and it is currently not clear if and how we would ever definitely answer it. It is one of those simple questions that open up a deep discussion about definitions and also philosophy.
There is a few important pieces of context that might not fully give an answer but are a great basis to think about it:
Our current cosmological model does an excellent job describing the universe down to the first fraction of a second. It is only a tiny fraction of a second where our theoretical models break down: physics becomes very different from anything we could build an experiment for here on earth and we also struggle to look back this far since our view is obscured by things like the Cosmic Microwave Background. But just because a tiny fraction of a second seems small on a human timescale that doesn't mean there isn't a lot of room for a lot to have happened back then that we don't yet know about: maybe our universe started from nothing, maybe it bounced back from a previous universe or started from within another. We don't even know if our universe is a finite size or infinite. All we are pretty confident about so far is that our universe has been expanding for many billions of years. When we extrapolate backward the entire part of the universe that we can observe today was converged to a single point around 14 billion years ago.
Time is a concept that works great in our daily life but there are limits to it. Even here on earth GPS satellites would be off within a day without corrections from special and general relativity. When falling into a black hole time gets stretched by an extreme amount. So when we say before the big bang, we can not just assume that something like "a second before the big bang happened" has to make sense. For example, the future can be infinite, a clock ticking forever, I don't need to imagine anything beyond that. In the same way in the early universe density diverges and the concept of time can stop existing as we reach time zero, there does not have to be a before.