Solar panels heat up. It's the law!

Don't forget entropy. Matter can absorb energy without necessarily radiating it immediately. Two objects can be the same temperature but of totally different energy. The easiest to understand is probably phase changes. Ice can melt into water, absorbing energy, without changing its temperature. Even stranger, ice can sometimes melt, still absorbing and retaining energy from its surroundings, while the temperature of the system actually drops (ie if you pour salt on ice).

Perhaps they use the heat very efficiently?

No dice. Conservation of energy and other thermodynamic laws say that [a] there are limits to efficiency, and [b] if you absorb energy you must re-radiate it. Among other things. There's no point in speculating about the blatantly unphysical. @Sharlin is right, it's just "Goddidit" dressed up.

Do any of them say it must be re-radiated omni-directionally as infra-red light?

Pretty much.

You can make it cooler than infra-red light (like microwaves), and you can irradiate for half a sphere, or even a bit less. But that's all.