Stirring the surface breaks the surface 'gunk', which gives the water a better chance to reoxyenate. It's just a matter of once or twice a day giving the top inch of the water a little stir. Otherwise a filmy layer can form and prevent any oxygenation. Specially if the bowl is in a brighter spot, where algae (the good and the bad ones) can flourish, and the waste product of these algae will create that filmy layer on the top faster than if the bowl is in a less bright spot.
The conditioner is important, since it removes chlorine and chloramine from the water - both of these are toxic to fish and can decrease their lifespan dramatically. You just need a tiny bit in the water, leave it for about 5 minutes before putting it in the bowl. Any pet supply store will have an aquarium water conditioner. For a small bowl like the one you have, a bottle will last you two or so years. You only need a tiny bit. When you put the dechlorinator in the water, you'll already see a swirling kind of motion happening inside the water, which is the interaction of the conditioner with the nasties in the water.
For the next week or week and a half, replace a cup or two of water of the tank each day. This leaves enough water in the bowl for the good bacteria, and gives enough fresh water for oxygenation and removal of the ickies. See if that perks your fish up again!
You might want to google 'cycling' and 'aquarium' to get a better idea of why to remove only a bit of water at a time, instead of replacing all or most of the water each week.