Hi, Dan!
To "talk my way out of" something means to try to find the right words to say in order to resolve a conflict that a person might have gotten him/herself into. For example:
"My boss called me into the office today because he wanted to punish me for skipping a day of work, so I tried to talk my way out of it." In this sentence, the person doesn't want his boss to punish him, so he tried to find the right words to say to avoid getting punished.
"Mom told me I had to finish my homework before I could play with my friends. I tried to talk my way out of it, but she wouldn't budge." In this situation, the person wants to play with his friends first, but his mother wouldn't let him, so he tried to talk to her to convince her to allow him to do it (which she ultimately didn't).
"He can talk his way out of any situation." This is a somewhat common phrase to describe someone who is so good with words that they could use their speaking skills to resolve any conflicts in their own favor.
I hope these help. Please let me know if my explanations are unclear or don't answer the question.