Originally Posted by R.D. Silverman

(1) The numbers being factored generally have special form that is
of interest to mathematicians. The factorizations can be useful in a number
of areas, e.g. raising the lower bound on odd perfect numbers, and
determining the multiplicative sub-groups of finite fields. Factorizations
have been used to construct LFSR's and pseudo RNG's.

(2) Not factoring per se, but knowing how hard it is allows us to choose
secure keysizes for public-key cryptographic algorithms.

(3) Noone sponsors it. It is a volunteer effort.

(4) What does anyone get out of any research project? It helps advance
the state-of-the-art.