Because B and C are one semitone apart, you have access to every chromatic note, (and many notes twice) if you have both of those diatonic scales.
The top half of the diagram above shows which notes are available in the B and C diatonic scales, and whether you push or pull to get them. I'm not sure, but I think different octaves might have different push/pull patterns - the principle is right though.
So, with a B/C melodeon, you can play in any key at all, by using both rows. For some notes, you have a choice of whether to push on one button, or pull on a different button.
The bottom part of the diagram shows D and G. You can see that there are lots of notes absent. D/G melodeons either have this limitation, or work around it with a third row or some extra buttons.
That's the "how", but what about the "why"?
To play a D major scale on a B/C instrument, the push/pull pattern is:
Pull, either, push, push, pull, either, pull, pull
... and to play a G major scale:
Push, pull, either, push, pull, either, push, push
Accordion/melodeon music is typically very rhythmical, and the rhythm is hugely influenced by the push/pull pattern. There is a distinct slur when changing note without changing direction. There is a distinct stop when changing direction.
I suspect that the push/pull patterns when playing on one row aren't a great fit with traditional Irish music -- whereas a lot of English morris tunes sound just right with the stops and slurs where single-row play puts them.
The B/C tuning allows a choice of push/pull patterns, including patterns that fit well with Irish rhythms and melodies.