I know that and is used to join two sentences or phrases. There are some places I’ve read that have
And is used at the beginning of a sentence. What are the occasions when this is done?
You start a sentence with a conjunction when you want to call a clause out for special emphasis. Examples:
We finally won a game against Notre Dame. And our best player wasn’t even in the game!
You can come up with $500 to pay the fine. Or you can spend 30 days in jail.
We were spent, bruised, broke, and bewildered. But at least we were home.
As you can see, each of those three examples could easily be rendered as single sentences. But breaking them up into a complete sentence and a sentence fragment beginning with a conjunction calls attention to the latter half. It elevates a second clause into an emphatic statement, which is somewhat more noticeable and memorable because of that.