To falsify is to alter or mangle something, like a message or document, in a way that distorts the meaning.
Since false things aren't true, to falsify something is to dishonestly change its meaning. If a friend tells you a story, and then you retell the story but change important facts, you falsified the story. If you lie in court — commit perjury — you're falsifying the facts. Faking someone else's signature is another type of falsifying, as is lying on your resume. Anytime you misrepresent the truth, you're falsifying.