What makes the Pillsbury canned biscuits puff up?
It could be achieved with almost any leavening, but I suspect it's baking soda or baking powder, and not yeast.

You'd just need to package it before the reaction is completed, and let it finish in the container.  This would create additional gas, which would pressurize the can.

As for why it doesn't continue to rise, it's because they can control how much leavening goes in, so they'll have a known amount of gas created.  Once that gas gets to one atmospheric pressure (roughly, depending on how close to sea level you are), it'll expand appropriately.

It's effectively the same sort of process that happens with fizzy drinks -- when you open the bottle or can, you allow the interior pressure to equalize with the surrounding, which causes the bubbles to escape.  The difference in that case is that the bubbles are dissolved in the liquid, and so may need some agitation or nucleation sites to escape.  But there's still a fixed amount of bubbling that could happen.