A brand new 1" pitch chain with no wear to the pins and rollers will measure exactly 1 inch per link, pin center to pin center, this is how one-inch pitch is defined. In an ideal word the pitch of a new chain and the profile of a brand new sprocket are designed to mesh so that each chain roller will exert a relatively equal force against the tooth surface it is in contact with.
At 12 1/8" measured over 12 links the chain has worn to the degree that the length has grown approximately 1% in length. This "stretch" is due to wear in the pins and plate holes rather than any actual stretching of the link plates.
Running a worn, elongated chain causes the chain to apply most of its force to fewer teeth with the result that the tooth surfaces will wear more quickly and the chain elongation will increase more rapidly
It is suggested that it is prudent to replace the chain when the elongation goes beyond 1% but some of this is more relevant to systems with aluminum chainrings as they are more easily worn than the thick steel inch pitch units on balloon bicycles. Still, sprocket and chain ring wear is an issue and installing a brand new chain on worn gears will cause the chain to elongate more rapidly for the same above reasons.
In the real world most of what we use is fairly worn out when we get it and while new components will ride and mesh more smoothly, even with some wear the systems are reasonably forgiving.
