The inverted subject seems to be the main problem throwing you off. Laurent is the subject of faire. One could also write the following with the same meaning.

Comme Laurent le fait remarquer en commentaire, […]

The second problem is the raising of the complement of remarquer to the left of faire.

Comme Laurent fait remarquer cela en commentaire, […]

This is common with the causative construction.

Paul fait manger des pommes aux enfants.
Paul les fait manger aux enfants.
Paul leur fait manger des pommes.
Paul les leur fait manger.

In a way, faire+verb acts as a compound, the inverted subject goes after the whole block on the right and the clitics come before on the left. Of course, it is not a real compound as adverbs can access the intermediate position as in:

Comme le fait souvent remarquer Laurent en commentaire, …