Explanation : Latching Relay Circuit
This looks very similar to a half bridge.

When both the IOs are at the same level, both the left and right circuits are providing the same potential to the relay.

When one of the IOs changes, the potential provided at that side of the relay changes.

The difference in these two potentials is what switches the latching relay.  It's just like driving a motor in forward/reverse.

Let's take a closer look at the right-hand side of the circuit.

When the IO pin is low, what is happening?

Well, Q4 and Q6 both have their bases pulled low.  This switches them off.  In doing so the pull-up resistor R2 pulls up the base of Q3, switching it on.  Thus, the power from the +12V rail is allowed to flow to the relay.

If the IO is set to high, what then?

Well, Q4 and Q6 are both switched on.  This connects their collectors to ground.  Q4 links the relay to ground, and Q6 links the base of Q3 to ground, switching it off.  This isolates the +12V from the relay so you don't get a nasty short.  so that side of the relay is now at ground potential.

Combine that with the same thing happening to the left side of the relay, and you can see how different combinations of IO can give different potentials at the relay terminals.

A | B || L | R | Vdiff
======================
0 | 0 || + | + |   0v    Nothing happens
0 | 1 || + | G | +12V    Relay switches to one position
1 | 0 || G | + | -12V    Relay switches to other position
1 | 1 || G | G |   0v    Nothing happens