Okay, here goes.
Back in the days of Abraham, there were things called covenants, contracts so strong and serious that to fail or break the covenant was punishable by death. It was way above and beyond a normal contract, even a marriage contract, in its seriousness. The parties entering into the covenant would perform a ritual to show everyone that they were entering into a convenant relationship, and that they were willing to die rather than fail or break the contract. They would take a live lamb or heifer and cut in in half, and let the blood run out and then walk with their bare feet in the blood and say, "This you will do to me if I fail to meet the terms of this covenant." It gave each person the power of life and death over the other.
Abraham was visited by God one day, while he was lamenting his lack of an heir. God promised him great things, the entire promised land and all in it, and Abraham knew that he would not live long enough to fully enjoy it. So he complained to God that it didn't matter what God promised, since he couldn't pass the promise down to an heir that would be able to take advantage of it. God told him that not only would he have an heir, his offspring would be as numerous as the stars in the sky. God made it a covenant, beyond a promise, to show how serious He was about carrying it out. He told Abraham to take a goat, a lamb, a heifer, a dove and a turtle dove and split them in half and let the blood run out. Then at nightfall, God appeared in his most mysterious, purest form, and walked through the blood, indicating that Abraham could kill Him if He failed to do what He said. Abraham had previously done the same, indicating that if he or his offspring should ever lose faith in the promise, God could walk through their blood.
So years went by, and not one of Abraham's offspring kept perfect faith in God. We have all sinned against Him. God had every right to kill us and walk through our blood. But instead, He came to earth and lived as a human, having perfect faith in the Father, and allowed men to kill Him as the punishment that we deserved from Him. It seems insane, but God allowed us to kill Him because we broke the covenant!
And that's how the Blood of the Lamb takes away our sins!