From the information you've given this sounds exactly like a poor water quality or living conditions issue. I would wager that since he left the conditions of the shop, he has slowly deteriorated. Good quality water will hugely reduce the chance of having issues with your fish.
What is the problem with my tank?
It's hard to say with the data that you've provided but here is my suggestion for getting to the bottom of your issues.
You'll first want to check the current quality of your water. By this, I mean checking the levels of the following:
- Ammonia (NH3)
- Nitrite (NO−2)
- Nitrate (NO3-)
- Chlorine (Cl)
You may want to consider getting a test kit like this to assist you in getting consistent readings.
Both Ammonia and Nitrite should be 0 and Nitrate should be as low as possible but under 40ppm is generally good enough.
If you have any ammonia your fish are suffering.
Other issues that might be contributing?
Check that you are keeping your fish in the correct conditions. Betta's needs are not much different to any other tropical fish. As a guide, check out this article which explains it all in great detail. Keep an eye on temperature, aquarium stocking, filtration etc.
If your tank is newly setup, make sure that it has been through a fishless cycle to prepare it for the fish. Here's a good guide about fishless cycling
Consistency is key! Keep your water changes regular (roughly 25% per week) so that you can maintain good water quality.
Overfeeding: this is a classic mistake for new fish keepers (including myself when I first started). A hungry fish is a healthy fish. Your fish should be very keen to eat and if they aren't, you are probably overfeeding.
Aquarium Size: As with most pets, the bigger habitat you can provide them, the better. If you're using a tank less than about 20 litres you'll probably want to look at upgrading if you are serious about keeping a fish in good health. Smaller tanks, often aimed at beginners are usually much harder to keep in a good condition. A small water volume often coupled with poor design does not make your job easier.
Specifically to this situation
- It sounds as though your fish may not live at this stage. With good water quality you can grow fins back really nicely but if you don't fix the water fast, your fish probably won't survive.
- My best advice is to read around as much as you can about setting up tanks, cycling and general fish care (This site has some brilliant questions and answer to browse).
- This youtube channel is an excellent, well respected source of fish related tutorials and 'How-To' videos. You will find a lot of useful information there.
- Finally, the site I linked to above carries a lot of good advice aimed at helping beginner get going.
Once you get the basics going, the whole fish-keeping experience will be much more rewarding.