I understand 凭 as a verb to mean rely. But I often read it and the seemingly related 凭着 in familiar contexts but still do not know what function it serves. Seeing example sentences doesn't make it any clearer. In what cases is it most used?
You are close. 凭着 expresses the idea of based on, according to or relying on, and when translating into English, prepositions such as on or by can often be used to convey this. You should be able to use it in situations where this concept needs to be expressed. As you suspected, its part of speech is best thought of as prepositional. 凭, when used in this sense, is equivalent to 凭着.
- 她在沙漠里走了一个星期, 就凭(着)太阳的方位来认路. (She walked in the desert for a week, using only the sun's position for directions.)
- 我们的猫凭(着)它的本能, 走了十五公里的路, 回到了我们以前住的房子. (Our cat went back to our previous house having walked fiften kilometres by instinct. )
- 你是凭(着)什么文凭来应聘这个职位的? (What certifications make you eligible for this job? / You are applying for this job based on what certifications?)
Keep in mind that translation is an art and you do not need to stick with the words that I used. Be flexible.