I understand : we move the first "2" left of Sigma, we replace "(h - i)" by "j" and "i" by "h - j"

We have j = h - i. Now let's look at the limits on the summation.

Lower limit of i: i = 0 leads to j = h - 0 = h

Upper limit of i: i = h - 1 leads to j = h - (h - 1) = 1

Now, some of my own confusion. We currently have

The problem is that the original summation has that h is 1 or greater. The second summation has that h is 1 or less. I cannot make any headway for this part.

This is merely re-writing the sum in a different order: 1 + 2 + 3 re-written is simply 3 + 2 + 1. So this is just the same series as above written backward.

Notice that we no longer have that pesky problem with the earlier summation limits. I can't explain that one either.

Notice that in the original summation we can replace i with any letter we want. i is called a "dummy index" for this reason. Notice that j is also a dummy index. So we can replace it with any letter. In this case we are going to ignore the definition of i above (because it's a dummy index) and set j = i (because it too is a dummy index.) So where ever you see a j in the summation you simply replace it with an i.

-Dan