To watch a card, click the star. If the star is filled on your card list or in the card view, you're watching the card. If it's not filled in, you're not.
Watching a card adds it to your Watching feed. It also causes you to be notified about card updates to that card.
Some people like to keep the number of cards they're watching very short, while others like to let it grow almost infinitely. It's all up to you and how you like to manage your work.