Why is the 19x19 board so much more common than other sizes?
A 5x5 board is interesting because beginners very quickly encounter all the basic rules of go - living, dying, territory, corners, sides.

A 9x9 board is interesting because the size is just right to develop fighting tactics.

13x13 lets you get into the mindset of occupying corners and linking tactics together.

17x17 might have been the next logical step, until people realized things were still too linear.. And then they brought it up to 19x19 ?

A fun fact is that early 19x19 used to have two black stones and two white stones, on the corner star points, making diagonals (d4 + q16, d16 + q4), which is nowadays considered to be an opening for fighting games. It may have been a first step towards a fully empty 19x19 board, or a more spiritual yin/yang reason, who knows.

19x19 board, people usually say, is interesting because it is the largest board size on which there are more points on the sides (under the third line) than in the center (above the fourth line). A 21x21 board has many more points in the center, and therefore the focus of the game would shift to fighting and living in the center, instead of trying to make territory on the sides -- since that would, obviously, just not be as beneficial anymore.