Explanation?
Leaving aside sophisticated theology for a moment, quite simply, Catholics pray as Christ instructed. Just look in the Bible.
During the Mass:
Priest says: "Let us pray in the words our Saviour taught us." (Or similar phrasing).
Then priest and people TOGETHER say the "Our Father" (just as Jesus taught the disciples, straight from scripture).
Then priest says: "Deliver us Lord from every evil and grant us peace in our day. In your mercy protect us from all anxiety, as we wait in joyful hope for the coming of our Saviour." (approximately)
Then the congregation (only) says "For Thine is the kingdom, etc...." It is a separate prayer.
Here's a somewhat unrelated but true story:
My mother went to grade school in Chicago in the 40's. Apparently in those days, class began not only with the Pledge of Allegiance, but the Our Father as well. Yes, this was a public school.
Before her first day, my Grandmother, a devout Catholic, told my Mom, "Now they are going to say the Our Father at the beginning of school, BUT, they are saying the Protestant version! So after you say "deliver us from evil," you stop! Stay silent for the rest of it!"
So God was then allowed in the classroom, but only the socially dominant WASP version.
And lead us not into temptation. But deliver us from evil. Amen.
The meaning of the "temptation" part here is twofold: not to put us to a test we have no strength to pass, and to grant us strength to overcome the temptation we do face.
The lacking a definite article "evil" is unfortunate and comes from Latin which lacks articles altogether. The Greek has a definite article, "liberate us from the evil". The preferred translation is "from the Evil one", that is from Satan. Translations to Slav languages, done directly from Greek render it that way: "liberate us from The Crooked One". The sequel said by the priest in the Latin tradition provides the necessary clarification in a different way: "Deliver us, oh Lord, from every evil". The point is that it is not some abstract evil that we want liberty from, but rather very concrete and numerous attacks of the Satan.