What is 'past null infinity'? For example, in the sentence "there is no incoming radiation at past null infinity".
 A: 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.  
A: 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.
A: 
I think it has something to do with the horizon of the universe
A: Incredibly basic answer from a non physicist who knows a really bright physicist. 
The entire Space time continuum can be described as “Past null infinity” to “future null infinity”. In other words, all of space time. 
To take one more stab at this, the farthest a photon of light could have come from is “past null infinity” and the farthest a photon could ever travel even if traveling for an infinite amount of time is “future null infinity”. 
So, as the lay person thinks about it. We are talking about the outer bounds of everything that ever was and everything that ever will be. 
Hopefully I didn’t just disappoint every scientist who ever spent time explaining things to me. 
