I was a Christian theistic evolutionist for years, I think I can give an accurate representation of the belief. I was on the liberal side, I guess I still am, so my explanation may be a tad skewed from the average TE.
Then give me a more elaborate, less oversimplified, explanation.
You look at the text itself, the theology, history, and commentary surrounding said passage. As far as Genesis 1 goes there is commentary calling for an allegorical interpretation since the beginnings of Christianity, most notably St. Augustine but he was not the first. It is also important to note that even when a historical interpretation was accepted oftentimes an allegorical interpretation was accepted too, the two aren't mutually exclusive.
As this Hebrew scholar advocates, and I agree, even the compilers of the Torah were not overly concerned with a literal interpretation. I linked this article previously but in case you didn't read it it is a good read on the subject from a knowledgeable person.
Genesis As Allegory - My Jewish Learning
As to how the non-literalness works TEs have differing ideas. Some of the more conservatives will say that evolution is true and each day represents some long period of time that kind of catalogues the progression of evolution. I think that is nonsense for various reasons, for one thing the order is all wrong. The conclusion I personally reached was that a "day" is best understood as a normal 24-hour day, I took a multilayered approach where it is a literal and allegorical story, yet the meaning is not in the literal but the underlying message; the literal story was simply the vehicle by which ancient Hebrews expressed their theology.
Perhaps a non-religious example might be a camp-fire story that talks about two campers wondering off at night and got eaten by a bear. The story is literal yet fiction, and the message is not that there were two boys that got eaten by a bear but that you shouldn't be an idiot and wander around the woods at night.
Similarly Genesis' message isn't that God created in 6-7 days, but that God did create and cares about his creation. It also expresses the thought that we are all one giant family (all descendants of Adam/humankind) which is used to justify one of the seven Noahide laws that prohibits murder. Also the Sabbath is clearly expressed in Genesis 1 and 2, thus the expression of the creation story in terms of days.
You seem to have a run-on sentence here. In any case I can't make sense of any of it.
The point is that non-literal interpretations of Genesis are nothing new, and have existed long before Darwin or modern science.
It is mythology and poetry, that isn't an opinion or an educated opinion, it is simply the terminology of religious literary scholars. "Mythology" in the context of religion has a technical use, basically any ancient story that is used to ponder questions of origins and our place in the cosmos. Poetry too doesn't have to be a modern rhyming poem, the Odyssey and Iliad are also poetry.
I have to go but mostly said what I wanted, until next time.