What does a greater measured redshift ($z$) directly correlate with for an observed object?

Answer

Greater distance and emission further back in cosmic time.

Since the expansion stretches the light during its journey, the farther away an object is, the longer its light has traveled, resulting in a greater expansion factor, or redshift.

What does a greater measured redshift ($z$) directly correlate with for an observed object?
DistancecosmologyObjectredshift