What does the " claw one’s way out" mean?
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Escape with difficulty.
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Thank you for your reply,but I still feel a little bit strange for your explanation.
Does “escape with difficulty” mean you bring difficulty with you and go away or mean you go away from difficulty?
Because I think “with” has the meaning of bringing something,and I regard person usually want to forget their suffering,so I think they would go away from difficulty rather than bring difficulty with them and go away.
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No; have difficulty in escaping. ‘With’ has other uses: dance with grace, swell with pride, shake with laughter, etc.
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So “with difficulty” equals to “difficultly”, right?
I get it ,thank you. This let me learn a lot. 
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Yes.
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To claw one’s way out means to use extreme effort to remove one’s self from a situation. Not necessarily “difficulty” for instance the action you need to take may not be difficult but more so time consuming.
If you fall into a hole, you can claw your way out. (physical example and what the idiom is based upon) Think of the action of digging your fingers into dirt and mud to pull yourself out of the hole. Like the claws of an animal.
Another example is to claw one’s self out of a situation that is not physical, such as a contractual problem, bankruptcy or a tax problem. You are in this situation and now you need to use extreme effort of get yourself out of the situation.
Your explanation is very detailed.
Thank you. 