Spells marked 'remains in play' stay in effect indefinitely when cast. They only come to an end when the target is slain, or else the caster is slain (or loses the spell for any reason), chooses to end the spell (which they can do at the start of any Magic phase) or leaves the battlefield. The Wizard can continue to cast other spells (but not the same remains in play spell in subsequent turns) as they require only a little concentration to keep a remains in play spell going.
If they are not dispelled immediately when they are cast, remains in play spells can be dispelled at any point during a subsequent Magic phase, using dispel dice as normal, or dice from the power pool as if they were dispel dice if it is your turn.
Remains in play spells do not retain the energy of their casting, and so a Wizard does not need to beat the original casting dice roll if attempting a dispel in subsequent turns, but rather the minimum casting value listed in the spell's description (so there is no need to make note of each spell's casting roll). If you are dispelling a boosted remains in play spell, you will need to beat the basic casting value of the spell (not the boosted casting value).