It means a contract, a relationship. If the law is broken, then the contract is broken. It was a relationship dependent on obedience of the law. It is not dependent on obedience of the law, but on Jesus dying for us. That is, the new relationship between God and man is dependent only on Jesus' love.
Blood is used in antiquity to seal a contract.
Moses sealed the first covenant with the blood of sacrificial animals. The Eucharistic meal is not just the Last Supper but also connects to the glorious banquet in heaven. This is hope for all who share at his table. One Main Point Jesus let the disciples participate in the preparation of his passover meal. Likewise, by giving us the gift of his Body, he has given us the food that we need to participate in his salvation plan.