TV tuner cards are used for various reasons. Their function is to convert a television signal into a format the computer can use. Here are some examples of use:
1. Watch TV anywhere you have your computer.
2. Recording your favorite TV shows for watching later.
3. Pausing live TV.
4. Recording game play from video game consoles (this is what I use them for).
5. Streaming video of game play on video game consoles to Livestream or something.
There may be more, but these are what I can think of off top of my head. I record all of my YouTube videos via TV tuner. There are some, like myself, who don't want to risk legal issues with emulators and thus turn to TV tuners for recording their game play with.