Answer: First of all, they aren't always. A blue ribbon only means first place in certain countries, including the United States and Australia. In the United Kingdom, first place ribbons are red and second place is blue.
In Holland, first place ribbons are often - but not always - orange.
Championship ribbons can be multiple colors, and are often larger in size than those given out for class victories.
As to why? It's tradition, pure and simple. I have no idea why those particular colors were chosen, why they're different in different places... (which sometimes causes confusion when people talk internationally). So, it's just tradition and it helps people know how well a horse and rider did when seen across the ring.
This Highland pony mare had a particularly good day at the Royal Highland Show in Scotland. Four red (winning in the UK) ribbons and a Championship sash. She looks like she knows it, too. Image source: Confuslefu via Wikimedia Commons.