Present Value Analysis
You need the last part because you want the present value. If you were to only use the 16 you‘d get to 4 years in the future, or you‘d be calculating the PV for 4 year operating cost of 2000 and concurrent 16 year cost of 2700. without the last part you‘d basically have two operating costs (one of 2000 and one of 2700) for the next four years and then 1  operating cost of 2700 for the next twelve years.

This is the way you can think of the cost in more granular terms

Purchasing cost: $15k$ no discount because purchased today.

1st four years overhead:
$
\sum^4_t{2'000\over(1+r)^t}
$ 

2nd 16 years of overhead: $ {\sum^{16}_t{2'000\over(1+r)^t}}\over(1+r)^4 $ 

The first part gives you the PV of the 16 periods, but because this would only start 4 years in the future you'd need to take the PV of the PV in a way.

The total of the cash flow is then the following:
$15'000 + \sum^4_t{2'000\over(1+r)^t} +{{\sum^{16}_t{2'000\over(1+r)^t}}\over(1+r)^4}$

Alternatively you could write
$15'000 + \sum^4_t{2'000\over(1+r)^t} +{{\sum^{20}_{t=4}{2'000\over(1+r)^t}}}$

Hope this helps.