Bridge is obsolete and used to connect to Ethernet so you don't want one.

You want a Sonos Amp. One will usually drive two pairs of speakers and you can add more Amps or use an impenance matching speaker controller.

Sort the wires out using a battery, 5hespewker will click when you have the correct two wires. Label them to avoid having to do that again.

There may be speaker Volume controls in the walls. Typically they are mounted at light switch height.

When using the battery scheme, a connected speaker will thump only at the instant of connection. You’ll probably need an assistant to help.

Looking at the wires in the picture, each of the fat white cables probably handles a pair of speakers. Likely this installation was done by a pro or an anal, skilled amateur. As I said above, I would not be surprised if there were Volume controls. The fat white four wire cables would connect to one side of a control and two smaller cables, each containing only two wires, connecting to a speaker.

I do have Ethernet wired throughout the house into every room - could I make use of it for a bridge? Or do I need an amp regardless?

BRIDGE is obsolete technology that, due to its age, often has reliability issues. BRIDGE was developed to enhance wireless coverage. Since you have wired networking available everywhere, BRIDGE will be ignored by other SONOS units. You cannot connect passive speakers to BRIDGE. You connect speakers to the SONOS AMP.

All of the currently manufactured SONOS hardware will ignore BRIDGE. Overall network performance will improve if BRIDGE is powered down.

The Bridge has a power and an Ethernet port, nothing to connect a speaker to, it is a networking device, not an audio device. Same for the newer but still obsolete Boost.

If you want to send sound to a speaker you need an Amp.