Simulating the universe in a lab

Physicists seem to be constantly simulating the universe in the lab. As a software developer who has written some basic simulations, I wonder how the heck they do this. As a layperson in physics, I wonder what they heck this looks like.

For instance in a NEW SCIENTIST article called "From Big Bang to Big Bounce", we have the following:

He was watchinga simulation of the universe rewind towards the big bang. Mostly the universe behaved as expected, becoming smaller and denser as the galaxies converged.

What does this really mean?  What was this guy really seeing?  Was it simply columns of numbers flying by, giving an array of values corresponding to time intervals?  Was there a graphical aspect to this?  I read about these kind of simulations fairly frequently.  Being able to "simulate the universe" seems like a remarkable feat, but given the size and complexity of the universe, I have a hard time imagining what a simulation would look like.

But I’m genuinely curious.  Does anyone out there know?