Originally the firstborns were supposed to be the kohanim. After the golden calf they lost that honor and it was given to Aharon and his descendants since they were from the tribe of Levi and the Leviim did not participate in the golden calf.
Because the firstborns are no longer the kohanim, they must be redeemed. Leviim and Kohanim do not need to be redeemed as they are still eligible to serve in the beis hamikdosh.