It means that he has moved two places up the order of ranking [of popularity] and is now in third place. The expression probably derives from a time when the display of this ranking was made by placing pieces of card showing the name of the song, etc. in to card holders arranged to show the positions of the competitors. A slot is a hollow groove in to which something can be inserted, for example, a coin or token to make payment for something, or a piece of card, plastic, metal, etc. stating, for example, the contents of a drawer, or file, the name of an employee on an office door, or indicating, on a display board, that an employee is at work.

In this sentence it is a noun meaning "an allotted place in an arrangement, sequence, ranking or schedule." Watch Me must have been in fifth-place, and has done something to advance to third-place (moving up in the rankings by two "slots"). For example, in a contest (like a car race) the contestants are ranked by a leaderboard such that the winner is in first-place, the runner-up is in second-place, and so on. A race car driver in fifth-place that passes two other cars can then be said to have "jumped two slots" in the race standings to third-place.