Why do we kiss the Mezuzah when passing through a doorway?

The reason is to demonstrate our love for this Mitzvah.
There is a source to this custom in the Talmud regarding the conversion of the Roman Caesar’s men (Avodah Zarah 11b). Onkelos the son of Klonimus converted to Judaism. The Caesar, his uncle sent a legion of Roman soldiers to retrieve him. However, he enticed them to Judaism by quoting verses, and they converted to Judaism.The Caesar sent a new legion but this time he instructed them not to enter into conversation with his nephew. As they dragged him from his home, Onkelos saw the Mezuzah affixed to his doorpost. He placed his hand upon it to kiss it and asked them, “Do you know what this is?” The soldiers could not contain their curiosity and answered, “You tell us” Onkelos explained to them, “It is the custom of the world that the king sits in the inside of the palace, and the guards protect him from the outside. However, with G‑d, His servants are inside their homes and He protects them from the outside, as the verse says (Psalms 121:8), “G‑d will protect your departure and your arrival from now and forever. They were inspired and they too converted to Judaism. After that, the Caesar ceased sending soldiers to bring him back.
A right-handed person should use his right hand and a left-handed person should use their left hand.
It is permitted to kiss the Mezuzah even before one has washed his or her hands in the Morning since the Mezuzah is covered in a case.