I tried to think about all the possible meanings of の but I really couldn't understand the sentence above. I don't know if の is connected with なんで, or if it's the explanatory の, or something else.
As @AeonAkechiさん hinted at in his/her comment to the OP,
is the same as
One of the things we like to do in colloquial Japanese is leave out particles like を, が and は (in their neutral forms), and even sometimes に. It can almost always be assumed from context what particles are missing, but it can be difficult if you're a new learner.
When we ask a question using a question word like 何, 誰, どこ, or いつ, we often use the pattern you found in the sentence you ask about, where we put the question word as near to the end of the clause as possible in order to emphasize it, though this isn't only done with questions.
The の here (using the OP's specific sentence) is used to nominalize the clause 「私があんた(を)連れ出した」, after which we request further information about it using 「なんで」.
Some more examples of this phenomenon, where the second example in each pair has a stronger sense of wanting to know the requested information (what is represented by the question word).
And one non-question where in the second case, the word at the end is being emphasized.
So in the case of the OP's sentence,
Why do you think I brought/took/lured(depending on context) you out?