This specific usage comes from a foundational paper on the philosophy of mind by Thomas Nagel titled "What is it like to be a bat?" and was made popular by Davi

This specific usage comes from a foundational paper on the philosophy of mind by Thomas Nagel titled "What is it like to be a bat?" and was made popular by David Chalmers in his writings on the "hard problem" of consciousness (https://en.wikipedia.org/wiki/Hard_problem_of_consciousness#Overview https://en.wikipedia.org/wiki/Hard_problem_of_consciousness#...). That origin is really the key to the phrase: Nagel's point was that there is no way of talking about subjective experience objectively. Rather, we always have to relate it to our own (think about how you might define subjective experience to someone else, for example). If you buy into Nagel's argument then "something that it is like" makes perfect sense when talking about subjective experience, because the only basis for comparison that speakers have for it is themselves.