The image is of you pointing an eye at something or someone. It means to watch carefully.
The idiom "keep an eye out for" refers to watching for something or someone. An example of a sentence using the idiom would be: Jeff should be arriving soon, so keep an eye out for him.
Keep watch on.
It is just an idiom and has no history.
About 400 BC in England , a
jump a lots
spying or looking someone all the time
This is not used as much as it once was, but people will know what you mean if you use it; it's not obsolete.
An excellent expression. To keep an eye, or to keep an eye on things means to maintain a watchful eye. You monitor what is happening with an appropriate zeal to the occasion. Another idiom would be, to keep an eye peeled. It means you are watching and protecting something, or someone.
This is not an idiom - it means exactly what it says. You should stay fit and healthy.
If something "catches" your eye, it has snagged your attention. You are focused on that thing for whatever reason.
Stay calm
To keep struggling and not give up.