"After Me" could be mistaken for "After You", or it could be mistaken for "Follow Me". The two moves do very different things.
After You targets a Pokemon and, if that target hasn't already moved this turn, they get to cut in line and be the immediate next Pokemon to move, regardless of speed order. One example where people have used this move is in the pairing of a Torkoal with Drought and a Lilligant with Chlorophyll; Torkoal is very slow but Lilligant is fast, especially because of the sun boost. It doesn't necessarily hit very hard though, so what Lilligant often uses its turn to do there is After You on Torkoal. This allows Torkoal to hit right away with Eruption, which is really strong at full health, but would be weaker if the opponents were able to hit it first and reduce its health before Torkoal got to take its turn. Because it's dependent on the After You Pokemon having high speed in order to accomplish anything, slow Pokemon like Clefairy don't have much point in using it.
Follow Me, on the other hand, is a move that is widely seen on Clefairy. It's a priority move, getting the jump on any damaging attack except Fake Out, Triage-boosted moves on Comfey, or Extreme Speed or First Impression from a faster user. Its effect is to turn the user (Clefairy in this case) into the center of attention for the rest of the turn, and what that means is that if a Pokemon wants to use a single-target move, but one of its opponents has the "center of attention" status, it's forced to target that slot instead of any other slot it originally chose to hit. For example, if you have a Pachirisu and Gyarados out, some players might want to use Electric moves to deal a bunch of damage to Gyarados if they don't know what you're up to. But if Pachirisu uses Follow Me, the move is forced to change target to Pachirisu instead of Gyarados, and it runs into Volt Absorb so that not only does it not deal any damage, it even causes Pachirisu to gain back some lost HP. In Clefairy's case, it's usually played as fully defensive with Eviolite, so while it doesn't have an ability that makes it immune to certain moves, it is at least better positioned to take hits than one of its more frail teammates, like Dragapult, might be, giving them more time to rumble through the opponent's team without a care. Furthermore, if an opponent tries to get around Follow Me by using spread moves, not only does the spread damage reduction apply to those moves, but Clefairy normally also has the ability Friend Guard to reduce the damage that the frail teammate has to take even further.