What does it mean to think philosophically about religion?
Since this is my area of specialization, I should be able to answer this question succinctly. However, philosophical thought is cognitive, placing ideas into categories. Philosophy is not necessarily affective, so why do people spend so much time thinking about various philosophies? Why, also, is philosophy sustained such long presence in academia? And, how does philosophy relate to religion?
Religion is like poetry about life, origins, and eternity. Religion moves people. Sometimes, on the one hand, religion can move people to misbehave, such as the extremists who use religion to foster hatred, misogyny, racism, and terrorism. On the other hand, religion can heal, comfort, drive people to share possessions and supply the material needs of other people. The poetry is explanatory and seeks meaning in life.
The tension is combining the two ideas: philosophy (cognitive) and religion (poetic and affective). This combination leads to a working answer: philosophy of religion is cognitively exploring that which is poetic and affective. The classic definition of philosophy of religion is ‘philosophy concerned with questions about religion’, e.g. the nature and existence of God, studying religious experience, texts, and vocabulary.
My definition might sound more clumsy but it tells what thinking philosophically about religion does. Philosophers explore with their heads, even though they cannot completely remove themselves from the subject they explore. Few, if any, people are ambivalent toward religion. Whether practitioners or not, people have opinions about religion, and when they study religion, they cannot divorce themselves from the epistemological reciprocity of their own past experiences with the subject matter.
Likewise, practitioners are not immune from cognitive explorations. People who are religious can asks philosophical questions. In other words, attending a Christian church does not mean that one cannot ask about the nature of God or studying religious texts. In fact, the most active philosophers of religion should be adherents because the people who have committed to religious practice have a reason to seek greater understanding of their practice.