The way I understand (two-port) transparent bridging is that the bridge simply forwards packets to another segment on the network. But since the MAC address of the ethernet interface is different to the wireless, how can they be transparently bridged? The router won't take packets with the ethernet interface's MAC address. It only knows about the wireless interface's MAC address. Doesn't the bridge need to change the packet's source MAC?
You seem to be confusing the different network layers. Routers route packets at layer-3, using the layer-3, e.g. IP, addresses, but bridging happens at layer-2, using layer-2, e.g. MAC, addresses. Bridging is done with a bridge, switch, or WAP, but not a router.
With an ethernet network, bridging is transparent. That means that the bridge doesn't modify the layer-2 frames when forwarding them. When you have two different LAN types, you need a translating bridge in order to translate the frames from one LAN type to another LAN type. This used to be common with ethernet/token ring bridges, and it is now common with ethernet/Wi-Fi bridges.
The question in your title, "How transparent bridging Wi-Fi with Ethernet works?" doesn't make sense. To bridge two different LAN types, you must use a translating bridge, not a transparent bridge. It so happens that Wi-Fi and ethernet actually use compatible 48-bit MAC addresses, so the MAC addresses can remain the same in the bridged frames.
A "self-learning MAC bridge" tracks the MAC addresses from the frames it receives - this way it nows, which frames need to be forwarded to which port.
WiFi bridging works the same way. MAC addresses follow the same pattern for 802.11 and 802.3, so the bridge just has to learn which addresses are on which side.