Offering lets you cast a creature with this ability any time you could cast an instant, but only by "sacrificing" a creature of the specified type (like Goblin Offering or Fox Offering). When you do, you sacrifice that creature as part of casting, and its mana value and colors reduce the cost of the Offering creature you're casting.
The big nuance: the sacrificed creature's full cost is subtracted—both generic and colored mana—but you still pay any remaining cost. You sacrifice during casting (as you announce the spell), not on resolution, so it happens before the spell is on the stack. Also, you're not required to use Offering; you can cast these creatures normally for their printed cost.
In Commander, Offering's flash-speed value shines for surprise blockers or end-of-turn plays, and it pairs well with sacrifice-fodder synergies and death triggers. It's a rare, niche mechanic mostly from the Kamigawa-flavored "Honden"-era cards.