The expression "I own this piece of meat" is used to describe the event of when a male lets others know in no uncertain terms that a female "belongs" to him.
Most instances are when the male feels threatened in some way by another male's presence, and so publicly gropes/manhandles his female partner to establish ownership.
This condition, stemming from low self-esteem, sexual inadequacy and physical inferiority, is often a manifestation of "small man syndrome".
When you meet a couple on the street, and there is little or no physical contact between the male and female. Immediately after the male sees you seeing them, he puts his arm around his girlfriend's neck or gropes her bum, and looks directly at you whilst doing said actions.
Inn this instance one can imagine the offender saying "I own this piece of meat". Saying this or similar manifests the attitude responsible for the actions.