Part of the description of the Slow spell says:
If the creature attempts to cast a spell with a casting time of 1 action, roll a d20. On an 11 or higher, the spell doesn't take effect until the creature's next turn, and the creature must use its action on that turn to complete the spell. If it can't, the spell is wasted.
The affected spellcaster casts a spell and is hindered by the Slow spell so that their spell does not go off until their next turn, but they save against the Slow spell at the end of their turn.