Animate Chains isn't a standard MTG keyword—it appears to be a made-up or misremembered ability, so there's no official rules text to cite. If you encountered this on a card, double-check the exact wording, as the actual mechanic is likely something else (possibly "Living Chains," a flavor name, or a typo).
If you're thinking of animating an artifact or enchantment, the relevant rules involve effects like Animate Artifact or March of the Machines, which turn noncreature permanents into creatures. The key nuance there: a permanent that becomes a creature this turn can't attack or use tap abilities unless it's been under your control since your most recent turn began (summoning sickness applies).
In Commander, animation effects matter for sneaking extra attackers or enabling combos, but verify the precise card text first—name accuracy is everything in MTG rulings.