What is the point to lots of the higher league players taking their scouting worker and mining from the enemy field? Is this just for BM? What is the purpose of this?
This is to disrupt mineral syncing. Experienced players will try to line up two of their workers to alternate mining from each of the closest mineral patches. So if they have 12 workers, 8 will be mining from close mineral patches and 4 from far mineral patches.
With your scouting worker, you can disrupt this by jumping on these close patches with good timing. This causes your opponent's workers to desycn from the patch and move to another, often starting a chain reaction. This can slow income heavily.
The goal isn't actually to steal minerals, it's to disrupt mining.
I've seen pro-level players use their scouting worker to try and disrupt the efficiency of the enemy's mining. Since only one worker at a time can mine any mineral patch, they will try to time their scouting mining so that it makes an enemy worker's AI believe that that patch is occupied or oversaturated, causing it to run off to mine somewhere else. They will usually do this on one of the closer mineral patches, as these are mined slightly faster.
It's unlikely that this tactic makes a significant impact on the game, even if executed perfectly. I think it mainly boils down to (1) the psychological effect or (2) just something to do with your extra APM in the very early game.
It's not only to disrupt syncing as the others have said but to completely prevent mining from one of the mineral fields.
While there is a worker mining a mineral patch no other workers can mine from it. So if you use your worker to mine from one of the mineral patches then they are losing 1/8 of their income (assuming they have at least 8 workers), which is quite a considerable effect.
To ensure that you stop them from using the patch as long as possible players will make sure that their harassing worker never finishes mining, they will click away to cancel mining and then right away click back onto the patch to continue mining. This means there is a very short amount of time for another worker to start mining, if left to the AI, it would never be able to mine from this patch.
In order to combat this the player getting harassed by this method will need to get one of their workers and spam click on the field that is being mined by the enemy so that there is a chance that they start mining when their opponent cancels for a moment.
Note you cannot use your workers to try to kill it because if they are doing it properly they will be behind your mineral patches, if it's an SCV it may be worth sending one worker to keep hitting it but if they have the APM to use this tactic they will likely be paying enough attention to not take much damage. Drones and probes will recover their hp so it's not worth it at all. It is best to just chase it off with your first military unit.
These sort of tactics are great for a slight early game advantage but it also helps to put your opponent off tilt, they may become flustered and start making silly mistakes that could cost them the game in the long run.