The idea is that it reflects the distance between two points if you only travel parallel to the axes. So to get from $(7,3)$ to $(4,8)$ you first travel 3 units left to $(4,3)$ then 5 up to $(4,8)$, for a total distance of eight. Hence the distance is the sum of the vertical distance and the horizontal distance.
It's meant to be evocative of the distance a taxicab would travel in a city where streets are in a grid pattern, so that you can only get places by going, say, east-west or north-south, rather than travelling directly towards them.