This phrase is attributed to Pope Leo XIII who reigned at the end of the 19th Century. He was also famous for initiating the prayers we used to say after Mass including the prayer to St. Michael the Archangel.
This phrase refers to spiritual rather than physical combat, although Catholics might be called upon at times to defend the faith physically, as they were during the Crusades. Mostly though it is the spiritual combat that is the most common.
We fight spiritually on 3 fronts – the World, the Flesh, and the Devil.
We have to fight a spiritual battle against the World because the World is constantly tempting us with distractions that pull us away from God. For example most of us have to work for a living and we can get caught up in the latest office politics, who is getting promoted (and perhaps we think we deserve it more than them). Or the latest movie or TV series that is coming out.
The Flesh refers to satisfying our sensual desires. We like to eat and drink, sometimes to excess. We are attracted to sexual pleasures. Each of us probably has at least one of these tendencies. While eating, drinking, sex, etc. are not bad in themselves, they can be if they become the main focus of our lives again drawing us away from God.
The Devil is probably the hardest one to combat as he knows our weaknesses and how to exploit them. We need God’s grace to combat any of these but especially this one.
The principal weapons for our combat is prayer for the graces to be strengthened and to trust in God. We can do none of this on our own but need to depend on God to pull us through.
St. Michael the Archangel,
defend us in battle.
Be our defense against the wickedness and snares of the Devil.
May God rebuke him, we humbly pray,
and do thou,
O Prince of the heavenly hosts,
by the power of God,
thrust into hell Satan,
and all the evil spirits,
who prowl about the world
seeking the ruin of souls. Amen.