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

"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"