What is meant by argue your own thesis?

A typical research paper looks something like this: The first paragraph sets the context for the thesis statement , which is generally the last sentence of the first paragraph. (I suppose it doesn't have to be there, but your readers will be looking for your thesis, so don't hide it or make it vague!) The rest of the paper should be devoted to presenting material in an order that works to both support your thesis and clarify your thoughts about the topic. In other words, your whole paper is geared around validating your thesis or proving it to be true.
And that's basically what your teacher means by "argue your thesis." While you're not engaged in an active debate, a good research paper must be clear and convincing enough so that even a person who doesn't agree with your thesis can understand why you've chosen to support it. That's how you argue a thesis. You back it up with facts and information.