If you have ever broken glass on a thick pile or shag carpet, or have cleaned it from car upholstery or floorboards, you can probably guess why it's so hard to get fiberglass out of fabric. Glass shards are thin and hard to see, and they often have jagged edges on multiple sides so they grab on when going into the fabric and grab on when you try to pull them out.
When fiberglass fibers get into clothing, they jut out in different directions and poke into the skin. Even after clothes are removed, the skin can be itchy, red and sensitive for hours or even days. Protective coveralls or cheap thrift clothing you buy with the intention of throwing away at the end of a project are options, but just as it's possible to gently get the fibers out of your skin, it is generally OK to keep your clothing and wear it again.
One tip for picking clothes to work in, though, is to choose loose shirts and pants so the fiberglass isn't pushing directly through the fabric into skin. A looser fit also will keep the glass splinters from getting pressed into the fabric as deeply because there will be less tautness between the clothes and skin.