When you want to send some bitcoins,  your wallet creates a "transaction" and sends it to a node on the Bitcoin network which then broadcasts it to other nodes.

When you initiate a Bitcoin, you must prove to every node in the Bitcoin network that you are authorised to spend those funds.

What is a Bitcoin transaction?

A transaction is just a message that coins information about the sender and recipient and the amount of BTC to be transferred (including a transaction fee).

So before sending out the message (the "transaction"), you're required to "sign it". All transactions need to be "signed" before they are submitted to the network. 

To do so, your wallet software does some fancy math magic to the transaction message and then after that, does some fancy math magic with something known as a "private key".

The results of all this math magic is a special piece of information called the "digital signature."

It is this "digital signature" that proves to the Bitcoin network that you are the true owner of the Bitcoins you wish to send.