Essentially, aging is just accumulated damage from these seven processes.
One of the seven processes is the accumulation of senescent cells. They clog up the works and also release chemicals that prevent normal cells from functioning at full effectiveness. Normally senescent cells kill themselves and are broken down by the body, but this process gets worse over time and senescent cells multiply.
The drug in this story targets and kills these senescent cells selectively. Effectively they are reducing the population of senescent cells in your body from the number you'd find in an 80 year old person so some lesser number (perhaps like a 20-, 30-, 40-, or 60-year old, but human studies would be required to say how effective it is).
But that's what aging reversal is. It's taking the seven biomarkers identified by SENS and reverting them from 85-year old levels to 25-year old levels. If all you're biomarkers are the same as a 25-year old, you are 25 in every sense except calendrical.
EDIT: typo