Sometimes I see people with AP builds on these typically AD characters.

To me:

AP Shaco: A part from planting boxes, hiding in bushes, and throwing his knife around, he doesn't seem to do too much. I feel like whenever I have one on my team I'm fighting in a 4v5.

AP Yi: I get that his Q does a killer amount of damage, but what's the point of a one trick pony?

I think I'm missing something because I see these builds all the time, but I just don't get it.

AP yi is a hilarious pubstomper because with his ult on, if he gets a kill all his cooldowns refresh so he can consecutively cast his alpha strike and get aces very easily. His meditate also makes him a very strong tank if the team doesn't interrupt him.

AP shaco is also gimmicky in that he relies on laying nests of jack in the boxes and waitin gfor people to facecheck and then get instagibbed. Also pretty funny but also pretty bad against competent players.

Ap shaco, although imo not as good as AD, can be a nice counter to AoE mages. Stick a clone in there, and it blows face off, and the AoE oriented mages can't prevent that.

Yi has a good ratio both on alpha strike and meditate, allowing him to tank tremendous amounts of damage given the right conditions. Also Alpha strike refreshes when he gets assits, so this means he can dash it out twice rapidly if a target goes down. Even with moderate amounts of AP thats twice 500 x 4 targets.

Last but not least AP Yi is still good at taking down turrets and has decent AD endgame, allowing a hybrid build if you so desire. Not a gamebreaker, but unlike classic mages he's not easily countered by MR.

AP Yi has a beastly lane if you learn exactly how Q behaves & reliably damage champions when targeting minions. As mentioned earlier, his teamfight depends on meditanking and resetting Q's cooldown through ultimate.

It's also very sensitive to the opposing team's composition.

I've personally never played Shaco so I cannot speak for him.

However, I am a big fan of Yi. It's always fun to play characters in ways that they are not the norm. (Like AD Sona) Both of his abilities that use AP scale very well. His Alpha strike can do a lot of damage, and when cast from a bush can easily catch an enemy off guard. The ability, Meditate, can easily heal you for 1500 hp with a single cast. This ability will allow you to farm almost indefinitely. Also, with all the Armor and MR it provides you can provide a tanky target for the enemy team to focus on while your team beats on them.

AP Shaco and Yi aren't very viable, however, they can be utilized when playing soloque against lower level players. Shaco simply spams Jack-In-The-Boxes in bushes, hoping to get lucky. Yi just owns with his Q, when low, runs away and heals like 150hp/sec!
Against good teams, i'd advise against it

AP Yi works if you can get a kill with your ulti on.