In physics, "escape velocity" is the speed (and direction) at which a body can escape a gravity well without further acceleration. That is, if you're moving fast enough you can escape from a gravity well without propulsion. For example, to escape from the Earth/Moon gravity well you need to be moving at 11.2 km/s. Of course, you can escape at any speed if you've got an engine, but if you want to, say, fire a bullet away from earth it would need to be moving 11.2 km/s to really get away (excluding air resistance); otherwise it will eventually fall back.
If that's clear, then consider the idea of "life extension escape velocity": the rate of life extension after which people will stop dying from old age. For example, since 1850 life expectancy in the United States has risen from 38.3 years to 74.8 years: an increase in life expenctancy of approximately 3 months per year. However, imagine a world in which life expectancy increases at a rate greater than 1 year per year... some people would still die, statistically, but the average age of death would increase faster than people actually get older. Death from old age would become very rare.
One year per year is the "life extension escape velocity". If and when medical advances ever reach that point, death from old age will be effectively eliminated. A lot of biologists (and others) have considered the issue and debated how society would change if people stopped dying, and Stanford professor Shripad Tuljapurkar has gotten some attention for making at least one obvious point: we'll have to raise the retirement age, something existing oldsters will fight against tooth and nail.