The player using In the Spotlight must announce their intent to use it immediately after a successful Mission Team Vote, and they must select the target player before any team member selects their Mission card. After the target has been chosen, each team member selects a Mission card and puts it on the table in front of them. The player who is the target of In the Spotlight then reveals their card to everyone. The leader then collects the other Mission cards as normal.
The other team members must select their Mission card before seeing the revealed card.
It's important to select the target of In the Spotlight before any team members have selected their Mission card, because it could affect the choice they make. For example, if a Spy selects a Fail card before In the Spotlight is played, they would have to pick up their card to change it. That would certainly be very revealing. The intent of In the Spotlight is that the target player knows they will have to reveal their card before making their choice.