It can be a literal area of higher ground, like a hill or mountain. You might hear this when there is a flood and people need to "get to the high ground."
It can mean moral superiority (real or imagined). To "take the high ground" in an argument means that you are either taking the morally correct side, or think you are taking the morally correct side of the argument. It can also mean that you are going to refrain from using dishonorable tactics in your arguments (like namecalling or personal attacks). For example:
Person 1: "You are an idiot!"
Person 2: "I'm going to take the moral high ground here and refuse to resort to such low tactics!"
It can mean to gain some sort of advantage. This derives from the military trying to capture and keep the high ground for tactical and strategic reasons.
"They really took the high ground with that new product of theirs."