What is 'past null infinity'?

Consider Minkowski space. "Past" means $t\to-\infty$, "infinity" means $r\to+\infty$, meaning infinitely far from your origin.

"Null" means "lightlike". "Lightlike" means you are going in that direction in the same way a photon would do, i.e. with $r/t=1$.

In other words, imagine a photon that crosses the origin at a certain time. If you follow its geodesic in the past direction, you'll reach past null infinity.

No incoming radiation at past null infinity is a sound border condition: I think it means, in very rough terms, no energy can come from outside the universe.

It corresponds to the $\scr{I}^-$ line of a Penrose diagram. "Null" means lightlike, "past lightlike" corresponds to the lower right boundary of Minkowski space, while "future lightlike" would be the upper right one.