They are proverbs, for want of a better word, and have developed and changed over time. For instance, a bird in the hand ... comes from the Bible, Old Testament, Ecclesiastes ix - A living dog is better than a dead lion.
They are all metaphors chosen to represent a situation in a more humerous and colourful way.
Where do they come from? Old wives, I guess.

the grass is always greener on the other side (of the pond), because americans paint their grass!!!!!

Meaning:
You use this proverb to say that the things other people have or their situations always look better than your own,
even when they are not really so.

It is often shortened to "The grass is greener on the other side," or even 'The grass is greener."



Example:
Don't compare your talents with others. The grass is always greener on the other side of the fence. Just do your best.

You might think you'd be happy if you were working in my company, but, well, the grass is greener on the other side.