Beef: Red on the outside, brown on the inside
The meat is brown on the inside not due to too much oxygen but due to a lack of oxygen.
Oxygen can pass through the packaging but not, generally, through the meat itself.  Thus, the interior of the meat runs out of oxygen faster than the exterior (which is still exposed to oxygen from the air) and browns for this reason.
I'll quote the full-explanation from the USDA, just for reference:

Why is pre-packaged ground beef red on the outside and sometimes dull, grayish-brown inside?
Oxygen from the air reacts with meat pigments to form a bright red color which is usually seen on the surface of meat purchased in the supermarket. The pigment responsible for the red color in meat is oxymyoglobin, a substance found in all warm-blooded animals. Fresh cut meat is purplish in color. The interior of the meat may be grayish brown due to lack of oxygen; however, if all the meat in the package has turned gray or brown, it may be beginning to spoil.