ActEd's "lightbulb boxes" in Chapter 10 prove quite helpful here...
Suppose you observe a life over the age interval (x+a, x+b), with 0 < a < b < 1.
Suppose the life survives
to age x+b. You've observed the life for (b-a) years. It contributes (b-a) to both the initial and central exposed to risk.
Now suppose the life dies
at age x+t, with a < t < b. You've observed the life for (t-a) years.
- It contributes (t-a) to the central exposed to risk...
- but it contributes (1-a) to the initial exposed to risk.
So the central exposed to risk is the actual time you spend observing lives. The initial exposed to risk has some sort of "adjustment" in respect of those lives observed to die.