She doesn't steal them, when a summoner spell is used by an enemy (And maybe an ally) they leave behind a bubble where the spell was used, Zoe can pick up these bubbles and hold them for a short time, allowing her to use the same summoner spell that bubble was formed from for a short time.
So basically enemy
Jinx and Sona see Zoe coming
Both Flash away
Both drop a yellow bubble, we shall call these flash bubbles.
Zoe flashes.
Zoe picks up the flash bubble Jinx dropped
Zoe flashes again
Zoe picks up the flash bubble Sona dropped
Zoe flashes again.
Sona uses Exhausts and drops an exhaust bubble.
Zoe picks up the exhaust bubble and casts exhaust on jinx.
Makes sense now? She is basically able to copy a summoner spell that was used by an enemy by picking up the bubbles that spells leave behind when used. I heard this also works on item activates but i'm not sure about that.