What does the verb "LOOK OUT FOR" mean? Is it British or American?

it's common in both british and american english.

it means:

1) to take care of or protect someone or something

EXAMPLE: My older brother always looked out for me when we were kids.

2) to be aware of and try to avoid something dangerous or unwanted

EXAMPLE: In the winter, drivers need to look out for icy patches on the roads.

Look out for is a phrase, not a verb (a verb is one word).

This is a command. Telling someone to do something (in this case, to look out for something).

It has various meanings.

It can be used as if someone were about to hit something (Look out for that rock!)

Or if you are telling someone to avoid someone (Look out for Sheila, she's evil).

I was born in London, England, and have lived twenty years in NYC and the American Middle West, and five years in Paraíba. I am not aware that "look out for" is any less common on one side of the Atlantic than the other. It means "to pay attention to what might happen or that you might encounter,, good or bad." When you go to Brazil, look our for good restaurants in São Paulo, for pickpockets on Ipanema beach, for the intricate patterns made in the mosaic sidewalk paving stones, for my sister who hopes to meet you on your arrival at GIG, for heavy rains and flooding in the South American midsummer, for strong cold winds from the Antarctic blowing onto the shore at Porto Alegre in winter, for words that do not quite mean what you might expect from knowing Spanish, for a warm welcome from most of the inhabitants, for a lot of bureaucracy, for investment opportunities, for misleadingly inconsistent spelling of surnames, for the punch line at the end of a Chico Anisio joke, for your young cousin who is likely to go wandering off on his own, for potholes on the highways, for ways to enjoy yourself......