But this rule goes beyond those animals killed specifically for a sacrifice. Any animal killed must be offered to the Lord in the prescribed manner (which in the case of a peace offering, God allows the one offering it to partake.) In fact, the only meat the Israelites could eat was that slaughtered at the tent of meeting and offered to the Lord. Perhaps this sheds some light on our tradition of giving thanks for our food and on the Israelites' complaints about eating only manna (even though clearly they had plenty of livestock from which they might have eaten -- but would be required to sacrifice it to the Lord to do so).
Think about the significance of putting God first each time you'd like to enjoy a meal. Not simply reciting a memorized expression of thanksgiving, but taking the time to carry your livestock to the tent of meeting, go through all the ritual associated with the offering (waiting your turn because there were hundreds of thousands of people doing the same thing). Nevermind the effort of simply preparing a meal in the wilderness compared to our current process!
What if we stopped before each meal and considered the gift that it is from God? How many people around the world don't have enough food to eat? How many don't have the means to simply drink a cup of cool, clean water? What if we looked for ways to "offer to God" a part of that meal? Perhaps by sharing a meal with a neighbor or a family in need.
What a great reminder that God has not only provided all that my family needs, He has provided all that His people need. But He hasn't necessarily distributed it evenly. He's trusted those of us who have more than we need to share with those who have less as the New Testament believers did in Acts 4:34-35: "There was not a needy person among them, for as many as were owners of lands or houses sold them and brought the proceeds of what was sold and laid it at the apostles' feet, and it was distributed to each as any had need. "