When people say they like something "big time," they're generally trying to make it clear how strong they feel about the subject. Putting this term at the end of a phrase is a way of adding emphasis, and it is used in a lot of other contexts as well. Generally speaking, when someone says he likes something "big time," it would usually be an attempt to express an especially over-sized preference, which would tend to be more significant than liking something "a lot" or liking something "pretty well."
One of the earlier uses of the phrase was in the context of becoming famous. For example, if someone were an actor in a local stage production, that person might long for a chance to make it to the big time. It is still commonly used in this way to describe differences between major and minor league sports, or an actor's rise from mediocrity to worldwide stardom.
It can also be used to talk about riches. If someone gets lucky in life and finds his way to major wealth, he might tell people that he had made it to the big time, and in this context, it could also be used for sarcasm. For example, someone might jokingly use the term because he got a slightly better-paying job or managed to win a consolation prize in a contest.
Most of the above examples describe good things, but the term can also be a bad thing. Somebody could have a big time fear or anger about something. Theoretically, someone could also use the term to describe fame that somebody gained from doing something notorious, like robbing a bank, and in that case, it could be regretfully stated that the individual finally became a big time criminal.
Many people believe that idioms are one of the key ways that languages evolve, and this phrase could be seen as a good example of that because it has developed so many wide-ranging uses. There are also a lot of other similar idioms that use the word "big" in front of another word to create emphasis, although many of them have more specific uses. For example, someone could be a "big wig" if he were in charge of a major company, and someone might have "bigger fish to fry" if he had crucial things to take care of and couldn't be bothered with the current task in front of him.