How can something happen when time does not exist?
General relativity is a local theory. That means it describes spacetime near the point you're looking at but it doesn't say anything about the large scale structure of spacetime.
Now this may seem unrelated to your question, but actually it's key to why we say that time started at the Big Bang. If we make a few apparently sensible assumptions about the universe we can solve the Einstein equation and get the FLRW metric. This metric allows us to start at our current position and trace back towards the Big Bang to see what happens. As we do this we are calculating a geodesic, which is simply the curve in spacetime followed by a freely moving object. The key point about this is that from some point on our geodesic, e.g. me sitting typing this, we use the metric to calculate the immediately preceeding points, then from there we use the metric to calculate even earlier points and so on back in time towards the Big Bang.
The problem is that as we calculate back towards the Big Bang the metric gets larger and larger, and at the moment of the Big Bang it becomes infinite. You can't do arithmetic with infinity. It might be fun to speculate what $\infty$ times 0 is, but when this sort of expression crops up in Physics it means we have to admit we can't calculate what's going on.
This is why you'll often hear it said that time started at the Big Bang. It's because we can't calculate backwards in time from that point. Now that doesn't necessarily mean there was no time before the Big Bang, it just means we have no way of calculating it from General Relativity. If you believe Loop Quantum Cosmology, this predicts that there was a bounce at the Big Bang, so we can follow geodesics back through the Big Bang and into an earlier universe. However this is highly speculative.
Incidentally, you get exactly the opposite effect if you fall into a black hole. If you launch yourself into a static black hole your geodesic is described by a metric called the Schwarzchild metric. The metric allows you to calculate your path towards the centre of the black hole (the singularity) but when you reach the center the metric becomes infinite and you can't calculate it any further. It's often said that time stops at the central singularity in a black hole.
Our daily-life experience of time is different from what physicans think that time is.
Time is just some kind of additional dimension. Imagine our universe was 2-dimensional in space like the surface of a table. Now think of time a third dimension, that goes vertical up from the table.
Every snapshot in time of this 2-dimensional universe is a 2-dimensional plane parallel to the tables surface. All snapshots together give a 3-dimensional space-time.
Our "real" universe, that has 3 dimensions in space, can be seen as a 4-dimensional space-time.
Particles moving through space when time goes by are transformed to lines in this space-time.
But when we think of time in this manner, as some kind of dimension, then we lose things line movement, motion, cause and effect. And the universe seems to be an absolutely deterministic place, where everything is in its place "from the beginning" beause in this universe there is nothing that we could "feel" as time. But this is not true.
Laws of nature are able to describe a space-time like I described here that would be experienced like separate space and time for intelligent beeings inside this universe. So this is not a contradiction of time as we experience it in our daily life.
So this universe has no edges and boundaries in all of its space-dimentions, so it is endless in space. But it can have a boundary in its time-dimension. Think of the tables surface as this time-edge. The universe is everything above this table. Nothing INSIDE this universe can be below this table, because "time" below this table does not exist.