To "put something on" your credit card means to pay for it with credit. When you use this phrase, it makes the expense sound like it's not that big. So it's useful for when you're talking to friends or acquaintances and don't want to seem like a cheapskate.
When you use this phrase, it's most common to say you are putting "this" or "it" on your card. "This" or "it" refers to the things that you're buying. For example, when you're buying a jacket, someone asks you if you're going to pay in cash, and you say:
No, I'm going to put it on my credit card.