The word "clincher" in an academic setting typically pertains to writing, and more specifically, to arguing.
According to Webster, a "clincher" is a decisive fact, argument, or remark. Think about it. A clincher clinches, or gets a hold of the audience so that they buy it, agree, or follow along. Typically, writers speak of "clinchers" in terms of finality, as in, the final clincher that got me to buy the car was that it came with a lifetime warranty on all parts and service.
If you have been asked to include a clincher in a recent essay, you are being asked to give one final reason to persuade your audience to agree with you. This means, save your best point for last. If you are being asked to identify the clincher in an argument (or essay), you should seek the the strongest point that is made, and expect to find it at the end.
I hope that helps.