Periods are the most straightforward of all the punctuation marks we'll look at. There is only one reason they are used: to separate complete sentences.
If you see a period in an answer choice, then you should check if it is separating two complete sentences. Start by checking the sentence to the left of the period; if it's not a complete sentence, then you can rule it out. If it is, then proceed to check the sentence to the right of the period. If it's not a complete sentence, then you can rule it out. If it is, then the answer choice is grammatically correct, and you've probably found the correct answer.
If the answer choice with the period is grammatically correct, then why is it only "probably" the correct answer? Well, occasionally punctuation questions are testing you on a bit more than just grammatical correctness. Sometimes they are also testing you on the appropriate use of a transition word or phrase. So the one additional thing you'll want to check is whether there is a transition word or phrase (which will be separated by a comma) either right before or after the period. For example, if the answer choice contains ", however" right before the period, or "However, ..." right after the period.
If there is no transition word/phrase adjacent to the period, then you can confidently choose the answer choice. But if there is a transition word/phrase adjacent to the period, then you'll want to check whether it's used correctly. Often in these cases, there will be another answer choice which also properly separates the two sentences (it might also use a period, or a semicolon, or possibly a colon), but the transition word/phrase will be on the other side of the punctuation.
If a sentence starts with a transition word or phrase right after the period, then the transition word indicates that the relationship of that sentence to the sentence before it. For example, if the sentence starts with "However,", then it indicates that the sentence should contrast with the sentence before it. If the transition word/phrase doesn't express the relationship correctly, then the answer choice is incorrect.
If a sentence ends with a transition word or phrase right before the period, then the relationship it expresses is less clear cut. For example, when a sentence ends in ", however" or ", though" it's often not indicating direct contrast with something else stated in the passage, but more of a contrast with what someone might have thought or assumed. It almost acts as a way of placing more emphasis on the sentence, and often indicates that the next sentence will elaborate on or exemplify what it's saying.
Let's look at an example to better drive home what we're talking about:
In the example above, the answer choices have semicolons rather than periods, but as far as we're concerned on the SAT, both serve as punctuation that can separate two complete sentences, so the same concept applies. Both answer choices C and D are grammatically correct, using a semicolon to separate two complete sentences. In choice D, the sentence after the semicolon begins with "However, ", which indicates that the sentence should contrast with the sentence before it. However, if we correctly interpret the two sentences, we'll notice that they are actually in agreement with each other. For this reason, choice D is incorrect. In choice C, the ", however" at the end of the sentence can be interpreted more as just a way of placing more emphasis on the sentence, teeing up that the next sentence might further demonstrate what it's saying. If we're determined to pinpoint some contrast that is indicated by the word "however" (since this word is generally used to indicate contrast), we can think of it as expressing contrast with what one might have assumed. Based on the first sentence in the passage, one might think that the neoclassical writers may have been the first to imitate the Greek and Roman epics, so the "however" is just expressing contrast with this theoretical assumption.