Fibers are strong because they are good at tension. This is true for fibers made of polymers: strings of compounds chemically bonded to each-other in sequence. Hair, cotton, and nylon are made of such polymers (keratin, cellulose, and nylon, respectively).

Many fibers have long molecules in them - spider silk, for example. Things like yarn or thread have lots of fibers in them, and the stronger yarns and threads have longer fibers in them.

Some fibers can stretch and stretch before they break. Other fibers don't stretch before they break. You just pull on them with more and more force and then they break. Stretchy fibers can do a lot of work, because work = force x distance [force times the distance they were pulled in stretching them].

Here's a link that says fibers are stronger than the bulk materials they're made of, because the bulk materials have defects in them, and the fibers don't:

fibers strong.