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I had lots of doubts about the paragraph. I wanted to clarify them with my teacher, so I went up to her/him. As usual, she gave very simple explanations, which helped me understand the paragraph clearly.
Dictionary says "clarify" means "to explain something to someone to make the topic easy to understand for them". So in short, "clarify" = "explain".
So in my sentence, I can understand this ("I wanted to clarify them with my teacher") as "I wanted to explain them to my teacher".
I think , in your sentence them refers to your doubts. By saying, I wanted to clarify them with my teacher, the student means, I wanted to make my doubts clear by consulting my teacher.
I hope, a teacher would take a look at it as well.
meaning of "to clarify them" 
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