I've asked this same question of multiple posters, as well as to give examples of spiritual truths gleaned and methods used, and the answers contain no information. One poster on this thread gave examples of what he meant after he commented that just because science can't detect spiritual things doesn't mean they aren't real, and then wrote, "If you sow lying, stealing and deceit will you end up receiving a blessing or a curse? If I am a generous person will I receive blessing or cursing? These are spiritual laws and realities." It's still not clear to me how that relates to science and reality, but I got a glimpse of what that one poster might mean spiritual - what I would call the wisdom gleaned from experience in a sufficiently decent and intelligent observer.
I've also asked the same question of the poster you're interacting with here. I'd address him directly, but he doesn't like that, and has blocked me. His answers are vague as you suggested, so it's hard to know if he's referring to anything specific that he can't articulate or whether he has no clear idea himself, but I've been reading his posts for a couple of years, and I think he's referring to the kind of learning we all do, but don't generally attach the word spiritual to.
What makes this area even more vague is the concepts of spirits as invisible beings, and when one attaches the supernatural to the process, there is a tendency to call it spiritual. And so, the claims for gods and angels and the realms of the afterlife are called spiritual truths, spiritual discernment, knowledge requiring the Holy Spirit. Now, it no longer resembles the kind of thinking a humanist has, and normal intellectual and moral growth and development are called spiritual growth.
For the record, I have a concise description of what I call authentic spiritual experiences. It is a sense of mystery, connection, awe, and gratitude that arises in various activities such as looking up at the night sky with an understanding of our connection to the stars, or a rapturous passage of music, or gardening. It's this experience that I believe theists are calling experiencing God or evidence for God. I had it myself in church settings when I was a Christian, and mistook it for the presence of the Holy Spirit. Then I was discharged from the military and returned to my home state, where I searched in vain for a congregation that was as full of the Spirit as my first (I became a Christian in the army), and realized that I had been in the hands of a charismatic pastor who created that feeling, not in the presence of the Spirit, which ought to have followed me to California. So, I'm pretty sure that that is what they are calling experiencing God after a few hymns and some singing and clapping that this is the day that the Lord has made, so be glad and rejoice in it.
By the way, that some interest and knowledge you have in religious history.