- Turn the chair upside down on a desk or on the floor.
- Use small scrub or tooth brush or mascara brush/wand, and clean the caster with soap water or alcohol. This will remove dusts and keep only hairs and threads that got tangled.
Level 1:
Remove the visible hairs using butter knife on the crevice of the caster on both sides. Kitchen Tweezers are very helpful here.

Level 2:
Use nail clipper/cutter's bottle opener. It will have pointed curve. Insert it into the fissure, Catch the hairs and pluck off the hairs. Keep revolving & rotating the caster strategically by other hand while doing this and this will help the tool to catch more hairs.

Level 3:
Simply cut a section of duct tape and then stick around the cover and pluck off. You can also cut a section of duct tape and fold it back around your razor knife so that it will be like a closed loop ready to stick, then insert it under the wheel cover and pluck off. Reason why Razor knife is because they are very thin.
TIPS:
Do not use sharp knives including razor knife. It will cut the hairs and make it difficult. But if the hairs are very stubborn and likely to tangled like a closed loop, cut the hairs with small manicure scissors or razor blade to break the closed loop.
If you are not satisfied after trying the mentioned workarounds, then I would recommend you to get a screw driver and remove the caster from the chair, then wash on the soap water and apply lubricants.
The example sample images are taken from Google search