I saw this word today, and thought straight away that its origins were pretty clear.
-scope is clearly related to watching or looking (periscope, telescope etc.) And horo- must be related to time (hour). And that’s the case.
It comes from the Greek horoskopos, made up of hora (hour, season, period of time) and skopos (watcher, that which is watched). There was one thing I was slightly wrong about though.
I assumed the word meant someone who looks forward in time, to predict someone’s future. But, it actually referred to someone who watches at the hour of another’s birth. It took me a few moments to figure out why that made sense: because a horoscope is based on the positions of the stars when someone is born (if I believed in horoscopes I might have realised that straight away).
This is why the original Greek word could also mean nativity, which I would have previously told you was crazy, but now makes complete sense!