The short answer is that we know Jesus was born free from Original Sin because it is written in the Bible. Hebrews 4:15 & John 8:46 are two examples; 1 Pet 2:22 quotes from an Old Testament prophesy, Isaiah 53, which says that He was born without sin. There is a longer answer, though.
The Bible gives us clues as to how Jesus' sinless nature fits perfectly with God's plan for our salvation. In the second presentation we talk about Typology, which means that events in the Old Testament prefigure, or present themselves again, in the New Testament. It's like a puzzle. If you look at the individual pieces, they may look distorted and not seem logical. But after the puzzle is completed, when you stand back and look at it from a distance, you can see the big picture.
In this case, God created Adam and Eve without Original Sin. A fallen angel, the devil, visits Eve and suggests that she disobey God. She does. Eve actively participates in Adam's disobedience of God, and the human race fell.
Think of our fallen race as a nasty knot. In order for mankind to be redeemed, repairing our relationship with God, the knot needs to be untied perfectly. How is this knot untied?
An angel visits Mary, who is born without Original Sin, and asks her to obey God. Mary agrees to obey God and Jesus is born. Jesus, who St. Paul calls the "new Adam" in the Bible (see Romans 5:14), is also born without Original Sin. Mary actively participates in Jesus' perfect obedience to God. Jesus' perfect obedience to God, submitting himself to torture and death on the cross, is what unties the knot and redeems humanity.
The fact that Jesus is born without Original Sin fits perfectly with God's plan for our salvation. The book Inside the Passion by Fr. John Bartunek puts it well: "This sinlessness was an essential characteristic of the Savior, prefigured by the Old Covenant requirement that sacrifices offered in reparation for the sin be animals "without blemish" according to Exodus 29:1"