A wedge is a tool used to split logs of wood. The thick end is hammered and the thin end steadily widens the opening until the log falls apart. The proverb means that a small beginning will lead, it is hoped (or more often feared), to something greater.
‘The building of just one house isn’t going to spoil the beauty of this stretch of countryside.’
‘But it’s only the thin end of the wedge. One house will be followed by another, and before we know where we are there’ll be scores of them.’