Is the wall box for a switch or a light?

Well the bare wires are easy, you connect those together and to the bare (or green) wire from the new fixture.

Does the switch that turned this fixture on turn on more than one light? If your husband only removed wire nuts for wires actually connected to the fixture (i.e., there weren't any connected together in the box that didn't connect directly to the light, chances are this feeds through to a second light (or fan). THen you just connect all the blacks together and all the whites (including the ones from the new light).

On the other hand, this may have been a switch loop. One black wire and one white wire from different cables going to the light and the other two were nutted together and not connected to the light.

If his memory isn't good, you can generally tell by carefully opening where the switch is. If there's only one cable (one black and white) coming to the switch it's probably the second case above. If the black wires going to the switch come from two different cables, it's likely the first.

There are 2 black wires coming from the switch itself