This is hard to truly eli5, so I'll have a go too, in case the others haven't cleared it up for you.

The spot on the moon that moves isn't a real thing, it's the effect of photons hitting the left side, then other photons hitting the right side. The 'reason' or 'cause' for those photons comes from earth very much at light speed. But the left side of the moon can't cause an effect in the right side, that fast. It just experiences a thing right before the right side experiences something similar.

Like if two cars drive from London to Manchester and Liverpool, arriving within seconds of each other. It doesn't mean you can drive from Manchester to Liverpool in seconds.

There's an SMBC I love on this: "The shadows of reality go as fast as they like." https://www.smbc-comics.com/comic/superluminal

Bonus: IIRC, any two events that are too close in time for light to travel from one to the other, can be viewed from a different "inertial reference frame" (someone else moving fast and analysing things with the same physics) as being the other way round. I.e. the right observer could see the right hand side of the moon get lit up before the left hand side. But the chap on earth wiggling the laser pointer is still wiggling it slower than the speed of light, so this observer would still see the laser pointer move from left to right. How does that work?