That is part of a Russound system, and it is the speaker distribution board.
Russound are one of the market leaders in distributed home audio.
I have no idea what system was in there, but it looks like an older system. It sounds as if you have at least impedance matching volume controls in the rooms, and may be something more complex.
It looks like it is a stereo system, with 4 sets of right and left speakers. I suspect the input impedance at the input of that board is 8 or 4 ohms, but I can not be certain of that.
I think you should contact Russound, or a Russound installer to tell you exactly what you have in your house. That way you won't blow up a new receiver and void the warranty.
Do not make any changes until you understand what you have. I suspect this was a complete Russound system, and is now missing the distribution amp.
This is definitely a two channel system and not a 5.1 stereo system. This was a background music system and NOT a home theater system, nor will you be able to adapt it as such.
If it is home theater you want, then forget that system, and build a 5.1 system in the room or rooms you want. Forget ceiling speakers for home theater.