You are infested but take no damage
A Swarm of Rot Grubs Bite action reads:
Bites. Melee Weapon Attack: +0 to hit, reach 0 ft., one creature in the swarm’s space. Hit: The target is infested by 1d4 rot grubs. At the start of each of the target’s turns, the target takes 1d6 piercing damage per rot grub infesting it.
Unlike in the case of Injury Based poisons the effect of Bite is not contingent on damage. This is especially clear in that when Bite occurs no damage is actually dealt.
You will, however, be immune to the piercing damage dealt by Bite. So you are infested by grubs but they cannot harm you.
How is being infested relevant?
Being infested matters in two basic ways.
Firstly it is relevant for role play/narrative reasons. There is a degree of body horror to having maggots inside you that could create an interesting scene. People who know you have grubs inside you might be disgusted and your character might be disturbed to feel them inside.
Secondly if you ever cease to be immune to non-magical piercing damage without having first removed the grubs (such as via Lesser Restoration) then you will start to take damage from the grubs. This could be relevant if you ever find a cure for your Lycanthropy.