Hello,
The effect reads : Once per opponent's turn (Quick Effect): You can detach 1 Xyz Material from this card; your opponent's cards and effects cannot be activated this turn.
That means that if you activate it in the standby phase (you can't really do it in the draw phase) , your opponent can chain effects to it and since the chain will resolve backwards, his cards will resolve first. If he doesn't chain to it the effect is active and he cannot activate cards and effects till the end of his turn.
If he had a card that destroys your Zexal but does not negate its effect, the effect is still in play, since the card does not need to stay on the field (it is not mentioned in the text). For example you activate your Zexal in the standby phase and he has a set Compulsory Evacuation device/Ghost Ogre & Snow Rabbit chainlink 1 Zexal chainlink 2 Compulse/Ghost Ogre so Compulse/Ghost Ogre resolve first removing Zexal 2nd thing that happens is Zexal's effect so your opponent cannot activate cards or effects till the end of the turn.
