Before there were motorways in the UK, A-roads were the highest level roads. And when they were originally numbered, the most important national roads were those given single numbers, i.e. A1, A2, through to A9.
Well, Edinburgh was the starting/ending point of four of these. Three of them, the A1, A7 and A8, still meet at the same junction. This is the junction of Waterloo Place, North Bridge and Princes Street respectively (and a fourth street, Leith Street, the A900).
The A9 also - originally - terminated in Edinburgh, though it split from the A8 in the suburb of Corstorphine. A long stretch of the A9 was over time declassified because Edinburgh Airport was built over part of it. Then part was downgraded to B-road and part renumbered since of the arrival of the M9 motorway running parallel. As a result, now the A9 only comes as far south as Falkirk, and doesn't even join with any other single-digit A-road.
But the sheer importance of Edinburgh to the road network is shown in these four roads converging on it, as well as some 2-digit A-roads and many other main roads. This importance should be immediately obvious if you look at Edinburgh on a road map.
Road distances to Edinburgh are measured to the A1/A7/A8 junction. This is where the GPO used to stand, and GPO's tended to be the point where road distances to a town or city are measured. A clue to this is given by a sign in the West End which says:
A8
City Centre 1
'City Centre', in road terms, being what was the GPO.