Let's use a smaller set of ranges to see how this works.

..........D..........E
1.......10.........10
2.......12.........15
3.......10.........20
4.......17.........20
5.......10.........30

=AVERAGE(IF(D$1:D$5=D3,E$1:E$5))

The IF function takes 3 arguments, a logical_test, a value_if_ true and a
value_if_ false. In the above formula the logical test is D$1:D$5=D3, the
value_if_true is E$1:E$5 and the value_if_false has been ommited so it
*defaults* to FALSE.

The logical_test will evaluate to an array of either TRUE or FALSE. Like
this:

D1 = D3 = TRUE
D2 = D3 = FALSE
D3 = D3 = TRUE
D4 = D3 = FALSE
D5 = D3 = TRUE

So, with the logical_test the corresponding value_if_true or the
value_if_false is then passed to the AVERAGE function and we get the average
of those values.. The value_if_true are the values in the range E1:E5 and
the value_if_false is the default FALSE. That would look like this: (T=TRUE,
F=FALSE)

D1 = D3 = T = E1 = 10
D2 = D3 = F = F   = F
D3 = D3 = T = E3 = 20
D4 = D3 = F = F   = F
D5 = D3 = T = E5 = 30

So, at this point the average function looks like this:

=AVERAGE({10,FALSE,20,FALSE,30}) = 20

AVERAGE ignores logical values (TRUE,FALSE) and text values that are
*elements of an array* so we get the average of 10, 20, and 30.

The formula you posted uses an empty TEXT string as the value_if_false
argument:

=AVERAGE(IF(D$7:D$28=D14,E$7:E$28,""))

So, this is what the average function would look like (based on the ranges
I've used in this example):

=AVERAGE({10,"",20,"",30}) = 20