Alright, let's dive into this one.
Think of a digital signature like a fancy, high-tech version of the old school handwritten signature. It's a way to prove that you're the rightful owner of something - in this case, Bitcoin.
When you send Bitcoin to someone else, you create a message (this is your transaction), and then you "sign" it with your private key.
This signature proves that it was really you who sent the message and not some scammer trying to pretend they're you.
Pretty neat stuff if I do say so myself!