Technically, no opportunity attack would be provoked

Note that opportunity attacks are provoked when you leave a creature's reach:

You can make an opportunity attack when a hostile creature that you can see moves out of your reach.

Meanwhile, the Boggle's Dimensional Rift feature states:

[...] While next to the rift, the boggle can see through it and is considered to be next to the destination as well [...]

Thus, the Boggle is considered to be in two places at once and because of this, when it goes through the Rift and later goes back through it the other way, nothing actually changes. The Boggle has not moved and the locations it is considered to be in have not changed.

Technically, thus means an opportunity attack is provoked when the Boggle moves from the other side of the Rift to a new location (assuming the creature on the other side can see the Boggle), but this usually won't do anything since only the Boggle can use the Rift (I say "usually" because things like War Caster exist).

That doesn't really make a ton of sense though

While the above interpretation is, to me, compliant with the actual rules, it doesn't make a ton of sense. If you can attack somebody who walks away from you before they actually leave your reach, you should similarly be able to attack a creature who Dimensionally Rifts away from you before they use the Rift (note, the Rift does not use movement in the game-sense of the word but it does involve moving in the English sense of the word and these can be contrasted with teleporting, which is instantaneous). At my own tables, I have not used a Boggle, but I would allow for opportunity attacks to be made against them. (I would also likely rule that the first time the Rift is used, it is such an unexpected attack that the character being attacked could not attack back).