Why is it most efficient to change orbit inclination while crossing the equator?

In the case of ANASIS-II, the situation is far more complicated than can be explained in ten seconds of livestream. Some general rules regarding plane changes:

1. You can only make a plane change at the point where your current orbit's plane intersects the target orbit's plane.
2. The faster you're going, the more fuel it takes to perform a plane change.
3. Because of how vector addition works, performing a plane change at the same time as you're doing an altitude-change burn is more efficient than performing the two separately.

The target orbit for ANASIS-II is a geostationary orbit. To get there from the Kennedy Space Center launchpad requires three maneuvers:

1. A plane change into an equatorial orbit, taking place over the equator.
2. An insertion burn into geostationary transfer orbit, taking place over the equator.
3. A circularization burn from geostationary transfer orbit into geostationary orbit, taking place over the equator.

Notice something? All three maneuvers need to take place over the equator, so maneuver 1 can be combined with either maneuver 2 or maneuver 3, and by rule 2 above, it's most efficient to combine it with 3, the circularization burn (a satellite at the high end of a transfer orbit is moving far slower than at the low end).

So why did the ANASIS-II launch combine the plane change with the insertion burn? Because it could. The Falcon 9 upper stage can carry more fuel than is needed to put ANASIS-II into a geostationary transfer orbit, so it used that extra fuel for the plane change. This reduces how much fuel ANASIS-II needs to spend to get into the target orbit, increasing how much it has left for station-keeping.

(If you're paying close attention, you'll notice that the "after" trajectory line of the maneuver isn't following the equator. A 28.5-degree plane change in low Earth orbit is expensive, and Falcon 9's upper stage can't carry enough fuel to do it. Still, even a partial change means a reduction in the amount of change needed during circularization.)

It's not just the most efficient way, it's the only way to achieve this particular target orbit.

As the other answers have pointed out, an orbital inclination change must occur at the so-called ascending/descending nodes, which are the two points in the orbit at which the current and target orbital planes intersect. Anytime a spacecraft moves from one orbit to another, the original and target orbits always share at least one point in common - it's where the burn occurred. If you want to move from an inclined orbit to an equatorial orbit, the burn must occur at one of two loci where the planes intersect, both of which are above the equator. If you don't perform the burn where the orbits intersect, you will never magically jump the distance between the two, and you will never reach the target orbit.

Adjusting inclination to reach an equatorial orbit must occur when above the equator - it's not just the most efficient way to do it, it's the only way to do it. You can potentially increase the efficiency of your maneuver by minimizing the spacecraft's speed and the resulting delta-v change, which requires moving into a larger orbit, performing the inclination change, and returning to the original orbit. But even if you move into a wider orbit, the inclination change still occurs when over the equator.

You can adjust inclination at either the ascending or descending node, and it'll be more efficient to do so at whichever one has a higher orbital altitude, since the spacecraft's velocity will be lower. So, it's possible that the mission in question chose to perform the burn at the ascending or descending node specifically to maximize efficiency. But the fact that the maneuver occurred over the equator has nothing to do with efficiency at all - it is in fact required if you're moving to an equatorial orbit with zero inclination.

To sum up, one cannot enter equatorial orbit from anywhere except above the equator.

A great aid to intuition is to remember one principle about orbit changes: if the engine is off, the orbiter always returns to same point one orbit later.

So for any orbit change, if you want to do only a short burn, it has to be at a point that is common for both the current orbit and the destination orbit. This applies to inclination changes, altitude changes and basically any orbit change. If the orbits do not have a common point, the change requires two burns and an intermediate orbit, such as Hohmann transfer orbit.

Like Russell's answer details, for a geostationary target orbit those common points are always above equator. For e.g. polar target orbit, the points would be somewhere else.