With an electron microscope and infrared and x-ray photoelectron spectroscopy. From the article:

The researchers counted the plastic particles by cutting open bags of tea, removing the tea, rinsing off any pieces that might have come off during cutting, and then steeping the bags in distilled water at 95 C. They then took a sample, let the water evaporate, and counted the plastic particles under an electron microscope, then extrapolated to get the amount in one cup. They also used other instruments to identify the type of plastic in each bag.

The researchers also repeated the experiment with uncut bags that still contained tea to make sure the cutting didn't cause the bags to shed, and with loose tea leaves, confirming that uncut bags shed microplastics too (although they were harder to count) and the plastic didn't originate from the tea itself.

The original study references the use of Fourier-transform infrared spectroscopy (FTIR) and X-ray photoelectron spectroscopy (XPS).

I would imagine at this scale, It's not so much counting particles, but weighing them. So you need to separate them out to do that.

Elements of different mass can be separated using centrifuges. Not 100% sure for this case, but I know that's how it's done for certain kinds of refinement.