Another way to explain it is this:
s + i = shi
So like Hyejong said, there's "si" which becomes "shi".
For the other ones, I understand the second "vowel bar" as meaning "+ y", which effectively turns them into "sha", "sho", etc. (but it's not the case with other consonants, only with the "s").
I don't know which explanation is easier to understand for you! ^^