The phrase Angst haben refers to being scared.

„Ich habe Angst, Frederick“, rief er.

"I am scared, Frederick," he called.

Caption 14, Piggeldy und Frederick: Das Kahnfahren

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What's interesting is that there are two prepositions that are commonly used with this phrase. When we use vor with Angst haben, it's a matter of being afraid of something or someone. 

Angst muss man vor ihnen keine haben.

You don't have to be scared of them.

Caption 3, Abenteuer Nordsee: Unter Riesenhaien und Tintenfischen

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Sie hat Angst vor der Prüfung, weil ihre Muttersprache nicht Deutsch ist.

She is afraid of the exam because her native language isn't German.

Caption 24, Bundesrepublik Deutschland: Einbürgerungstest

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But it's also possible to use um with Angst haben, and this has different nuances. It can actually be used to describe being afraid on someone's behalf, or being afraid "for" something rather than "of" something. 

Der Weltmeister bekam wohl langsam aber sicher Angst um mich.

The world champion was slowly but surely getting scared for me.

Caption 39, Rheinmain im Blick: Sportreporter als Judoka

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Auf dem Nachhauseweg verirrte er sich in einem furchtbaren Schneesturm und hatte Angst um sein Leben.

On the way home, he lost his way in a terrible snowstorm and feared for his life.

Captions 19-20, Märchen - Sagenhaft: Die Schöne und das Biest

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Further Learning
Try to create your own sentences with Angst haben and both prepositions. Many examples can be found on Yabla German.