Your air conditioner works by first compressing “freon,” (it’s not really freon in your car because it was built after they banned the refrigerant used in older cars, but people still call it that) and then expanding it. When gasses expand, they get colder. The air conditioner blows air over the part that expands the freon, which cools the air down, and gives you air conditioning. If your system has a small leak, then eventually the freon will leak out, there won’t be any gas to expand, and you won’t get cooling.
Recharging the freon just means adding freon back into the system. You put the dye in it so that if you lose the freon again in a short amount of time, you can shine an ultraviolet light on the air conditioner parts - when you see a glowy trail, you’ve found the leak and can fix it.
You can pick up an air conditioner recharge hose from an autoparts store. Should be around 15 bucks. It’ll have a pressure gauge on it. You hook it to your charging nipple and it will tell you what pressure your freon system is at - this will let you verify whether or not the shop is correct that you need to recharge it. At that point, if you do indeed need to recharge it, you can either do it yourself with a can of refrigerant from the auto parts store (be sure to just get plain refrigerant, without any “subzero” additives or stop leak additives, and wear gloves to avoid freezing your fingers if some of the refrigerant leaks out while you’re screwing the can onto the recharge hose) and see how long it lasts, or you can have the shop do it for you.
I suspect the shop may be wrong in its diagnosis if your AC was working well before you drove through the water, and stopped immediately after you drove through it. I suspect you may have shorted out the electronics that control your AC compressor’s clutch - if the compressor won’t engage, then you can’t compress the gas, which means you can’t expand it to make it get cold.