Alex told Katie to go away after knowing what she is, Katie packing her things without explaining to Jo. Jo asking her.
Jo: Look, Katie, if you run now, you're taking the easy way out.
Katie: The easy way? You think this is easy for me?
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Sign up to join this communityThe "easy way out" usually refers to an action that is considered mentally "easier" to do, when compared to an alternative.
In your example, Jo is basically saying that it's easier for Katie to just run without explaining, versus the harder option of staying and confronting Alex.
It will often refer to this specific case, actually: "easy way out" meaning to run away from something rather than confront it.