What does "pay attention to the weather" mean? Does it mean "look carefully"?

Weather eye: Pay close attention to the weather and be alert to storms and changeable weather. A sailor will say "keep your weather eyes open" or "keep your eyes on the horizon"

Various online dictionaries give a similar explanation, probably: eyes that are good at observing the weather.

The original sentence "keep alert to the horizon" should be translated literally: always pay attention to the horizon.

I hope it helps you!