What is meant by 'backing up information'?

Depends on the context of the phrase.

In the use of computers, backing up information, means to save it in more than one place incase something happens to one type of storage you used (i.e. hard drive crashes)

Or if another context such as saying, "I am a great cook", backing up your information, could mean to show proof. which in this case would be to offer a dish.