I just had to fix a couple of cell references. Sorry.

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I spent some time studying this myself, and I think I can explain it to you.
Though I could not have come up with it on my own. :)

The SMALL function, once you copy it down, simply sorts the rankings from
lowest to highest. See the help for the SMALL function. The ROW function
call (A1 in the first cell) returns "1" (which is the row containing the
cell A1). This tells SMALL to find the 1st lowest value. In the next row,
the ROW(A2) will give you the second lowest rank, and so on down the line.
In this example, this column of ranks is the range C2:C6 on sheet 2

The ROW function is a fancy way to get the numbers 1 thru 5 as you copy the
formula down.

The IF/ROW functions will return an array to feed into LARGE. As array
functions they will cycle through the whole range (C2:C6) looking for the
value in C2. For those rows that match C2, it will return the line number
in the table (not the row number on the worksheet).

With this data:
Student Test Grade Rank
Meep 100% 1
Batman 90% 4
Superman 95% 2
Willy Wonka 95% 2
Veruca Salt 43% 5

If you put the whole INDEX formula in cell A2 on sheet 2, the IF/ROW call
will return this array (internally):
1
false
false
false
false

Since "false" evalutes to zero, the "1" (which means the first row of data)
is the largest value in this array. The COUNTIF will count only one
occurance of rank = 1. The LARGE function will find the largest value in
the array - which is 1 - and pass this to the INDEX function, which will
pick off the name from the first row of data. When you copy this formula to
cell B2 on sheet 2, it will do the same work and pick off the corresponding
score.

So far, so good?

When you copy the INDEX formula to cell A3 on sheet 2, you are now looking
for people and scores for rank = 2. The IF/ROW functions return the
following array:
false
false
3
4
false

and COUNTIF reports there are 2 occurrances of rank =2.

LARGE will pick off the 2nd largest (from COUNTIF) and return the 3 to the
INDEX function, picking off the 3rd row of data.

When you copy to cell A4 on sheet 2, you still want rank = 2 (since there is
a tie for 2nd place). IF/ROW returns:
false
false
3
4
false

BUT - the range for COUNTIF changes and ignores the rows above the current
row, so COUNTIF returns 1. LARGE picks off the 1st largest value, and
passes 4 to the INDEX function, picking off row 4 of data.

A lot of this is done internally due to the way Excel handles array
functions, so you won't see most of this at work. When there are ties in
rank, this will keep the tied people in the same relative order as they were
in the original list.

Hope this helps.