Rushing is when a spy immediately moves to accomplish a mission right as a game begins. The purpose of doing this is to get a mission done during the Time of Chaos, when it will be more difficult for the sniper to notice a tell. However, as the spy will not have had any time to plan their moves, they are more likely to commit errors. Missions with hard tells are particularly desirable to rush, especially Bug Ambassador and Purloin Guest List. Swap Statue is a more tricky case. If the spy can do it quickly enough, they may manage to swap it before the sniper has memorized it, meaning that they'll never notice that the swap occurred. On the other hand, since many snipers memorize statues first thing at the start of the game, a spy who rushes the swap may end up swapping it as the sniper's looking right at it to memorize it.
Rushing can also be very effective to add time. If the Spy instantly moves to the correct location and adds time (by looking at the watch), the sniper may not be paying attention and will allow more time to complete the missions.