After Effects doesn't have variable width strokes so trying to import them from Illustrator will just expand the stroke. If you're trying to animate a variable width stroke then you'll need to animate a regular stroke and use the outlined variable width stroke as a mask.
Unfortunately with a single intersecting path then you'll end up with something like this:
There's not much of a way around that while keeping it as a single stroke (you could cleverly mask the overlaps, but that's not very practical). The easiest solution is to separate the paths so each overlap is its own distinct path (with its own mask too)...
I'll walk you through how I did this. The explanation looks long and complicated, but it actually only took me ~15 minutes once I knew what I was doing.
In Illustrator, you first need a path for the stroke animation, and an outlined stroke for the mask. The path for the stroke needs to have a basic appearance to import in to After Effects correctly; no brushes or width profiles. It can have a basic stroke but nothing else (and you can change the stroke once we're in After Effects anyway). The outlined stroke to use as the mask just needs to be an outlined version of your variable width path.
You should have something like this (viewed in outline mode):
Still in Illustrator, separate the paths so that no path intersects itself (in this case you need three paths). You need to do this with both the stroked path and the mask paths.
This looks a bit complex but it isn't really, you just cut both the path and mask in three and place everything in its own layer (After Effects needs everything in its own layer to import properly). The overlaps are just to prevent rendering artefacts where the paths meet.
Import the AI file in to After Effects.
Convert the first stroke path layer to a shape layer (Right click layer → Create Shapes from Vector Layer), that should give you a new shape layer with a single path — you can then delete the original AI layer.
Set the stroke width wide enough to completely fill the mask and add a Trim Paths from the "add" button on the "Contents" property of the layer.
Then all you need to do is animate the "start" and "end" of the trim paths.
Something like this:
Repeat for the other path segments and you should have something like this:
That doesn't loop properly, but you can just duplicate the needed keyframes and loop at the appropriate place:
All that's left to do then is mask each stroke. You could use a regular mask but it's easier to just use the layers you already have as track mattes (track mattes are basically opacity masks, rather than vector masks).
Just make sure each mask layer is directly above the relevant stroke segment in the timeline and choose "Alpha Matte" from the track matte dropdown on each stroke layer: