Why do people share their PGP Keys and how to use it?

Solution 1:

Why do people share their PGP Keys?

Because without giving someone your public key, they can't encrypt data that your private key can decrypt or confirm that your digital signature was created by your private key.

Wikipedia has what appears to be a decent article on the subject.

How can we sent an authenticated mail using a PGP Key?

Usually by using an email client that supports PGP/GPG. e.g. Enigmail for Thunderbird

Solution 2:

You can think of a PGP key as two locks on either side of a safe.

People with your public key can unlock one side of the safe and place items in it. The only person who can then take them out of the safe is you with your private key.

You give people your public key so they can send you emails only you can read.

Plus - Any messages sent to people encrypted with your private key can only be read with your public key. This is a great way of proving that you are the person who sent a specific message.