In generations before Gen 5:
>In Generations 1-4, the opponent's defense stat is temporarily halved when calculating the damage. This gives Explosion an effective power of 500.
Explosion indeed does give STAB, the only Pokemon getting this boost though is the Lickilicky chain. As the defense-halving is factored after STAB, the effective base power will be:
250 (original base power) x 1.5 (STAB) x 2 (halving of defense)
= 375 x 2
750 will be the base power of Explosion when a Lickilicky uses it.
In generations after Gen 5:
Since Generation 5, the base power of Explosion has been set to 250 without any kind of defense halving. So a Lickilicky using Explosion will still have 375 base power (factoring STAB).
Hope I helped!