The basic idea of the bus bridge is that bus operations slaved on one side are mastered on the other. The bridge need not be symmetric: speeds and data widths may be different on each side.
A bus bridge connects together two busses that are potentially able to operate independently when traffic is not crossing. However, in some circumstances, especially when bridging down to a slower bus, there may be no initiator on the other side, so it never actually operates independently.
The bridge need not support a flat address space: addresses seen on one side may be totally re-organised when viewed on the other side or un-addressable. However, for debugging and test purposes, it is generally helpful to maintain a flat address space and to implement paths that are not likely to be used in normal operation.
A bus bridge might implement write posting using an internal FIFO. However it will generally block when reading. In another LG we cover networks on a chip that go further in that respect.
Note, the 'busses' on each side use multiplexors and not tri-states on a SoC. The multiplexors are different from bus bridges since they do not provide spatial reuse of bandwidth.
With a bus bridge, system bandwidth ranges from 1.0 to 2.0 bus bandwidth: inverse proportion to bridge crossing cycles.