Car speakers only need the speaker wires going to them.

Home speakers *usually* only need the speaker wires going to them. Unless they're self-powered but wall-mount speakers usually aren't self-powered.

In the photo you show, that "board" attached to the speaker is just the crossover for the tweeter/woofer. It's entirely passive electronics and doesn't require a separate power supply. So the speaker pictured only needs the speaker wires going to it.

The only issue you'd have with putting car speakers into a home wall location is that most home speakers are a different ohm impedance than cars speakers. I don't remember which is which, but one is usually 4ohm and one is usually 8ohm impedance if I recall correctly. So if you want to use car speakers in your walls, make sure you know what their impedance is, and whether your home stereo can handle that.

At the very least, putting car speakers in as your rear surround speakers will result in large volume mismatch between your front and rear speakers, possibly a large enough difference that you won't be able to compensate for it in your amp's speaker balance adjustment settings.