Going back to the Galio update, petricite was the demacian Material that blocked magic.
It’s proven that just it being present can dispel/absorb magic with the Galio story. So why can sylas do his shit?
His abilities are all based on his chains sure, but how can he steal someone’s magical abilities when magic is nullified around him? How powerful of a spell caster is he? Why can he use magic while being locked with petricite?
Piggybacking off of MagicFlyingLlama's comment, they do actually explain the deal with petricite in Sylas' bio;
Eventually, through careful manipulation, he convinced the girl to smuggle a forbidden tome into his cell—the original writings of the great sculptor Durand, detailing his work with petricite.
The work revealed the secrets of the stone to Sylas. It was the foundation of Demacia’s defenses against harmful sorcery, but he came to see that it did not suppress magic, but absorb it.
And if the power was held within the petricite, Sylas wondered, could he release it…?
All he needed was a source of magic. A source like Lux.
Between petricite's magic-absorbing properties, Sylas' knowledge of petricite, and his natural aptitude for magic, it isn't far-fetched for him to be stealing people's magic with his shackles.
Galio also uses magic, keep in mind.
It absorbs magic, but nothing suggests it destroys or disrupts it. Looks like he has a way to either fully saturate it, or siphon off what it absorbs.
I always had a theory that Garen was magically inclined. Petricite after all needs magic to trigger its anti magic property. Galio cant move with out magic near him via lore. So I figured if Garen Is able to at will make his sword and armor function with anti magic, much less Summon Giant freaking swords out of thin air, he might be magically inclined.
Well more lore is out and its clear that Petricite stores magic. Looks like it eats manna, It does not destroy it. So i'm still entertaining the notion that Garen might have magic in his blood. Or failing that, he innately learned how to release the MP his armor and sword has stored.
Anyhow Sylas has the innate ability as a mage is to copy cat magic he comes in contact with. So he uses the chains to eat spells, then he mimics it by contacting the stored Magic in the chains.
When he tried to shield the child from the mageseekers, he accidentally brushed against her skin. The girl’s magic rushed through Sylas’s body—but rather than killing him, it shot forth from his hands in raw, uncontrolled bursts
