Although God provided evidence of his existence (even with people being aware of God), people ignore it, downplay it, or pay no heed and do all sorts of not-very-good actions as a result. What happens as a result? Because of not paying heed to God, God "gave them up to uncleanness" (1:24). The phrasing means that God lets them screw themselves over via uncleanness and other bad stuff -- likely so they can see that sin ruins lives. Basically, the famous calculator-mantra of "garbage in garbage out" is used. The people put in garbage by "exchang[ing] the truth of God for the lie" (1:25), and garbage almost always comes out in response.
One of the outputs is "women exchang[ing] the natural use for what is against nature" and "men with men committing what is shameful" (1:26-27). Time to talk about this! It's heavily implied that Paul is referring to reproductive actions (i.e.: rape) because the Greek word for "use" (Strong 5540) has a connotation of reproductives. From the wording of "nature" (as in biology), it's also implied that 1:27 is referring to the specific act of homosexual intercourse.
So that's equivalent to the modern day notion of homosexuality, right? There is another confounding factor -- the context of the homosexual intercourse. In Ancient Rome, much of the forms come in the form of pederasty ... which is different from the modern day attitude of intercourse based on mutual covenant similar to marriage. Still, there's an argument that the homosexuality is all forms of it ... and I'm still digging on it.
But for one, I'm not one to actively recommend people doing homosexual intercourse -- hear me out on this one. Biologically, reproductives are used for reproducing (along with, for some reason, pleasure, which incentivizes reproduction). In many animals (including humans), the two main biological sexes have complementary parts (for various biological reasons) -- for the case of humans, the sperm meets the egg to form another human. Various sexually transmitted diseases (STDs) can be spread via reproductives (i.e.: HIV/AIDS) -- and the risks heavily increase with more than one person. However, homosexual intercourse can not do reproduction (missing sperm or egg) -- but STD risks are still there (hence the effects that Paul talks about). Considering that God knows his creation very well (especially after the Fall), it's very likely that God knows all that and put in the reproductive rules so we have the best chances of staying healthy (not to mention that diseases screw over people too).
By the way, keep in mind that Paul is specifically talking about homosexual intercourse. As for the "burned in their lust" part in 1:27, the phrasing of "burned" as well as how lust is desire overfocus by definition indicate that the inner attitudes are specifically uncontrolled impulses (especially since it led to the intercourse). In other words, this is NOT referring those who simply have general same-sex attraction or even general love between same-sex duos (only a subset, and note from the definition that attraction is NOT necessarily regarding sex). After all, there are gay, lesbian, and straight people who have restrained from intercourse before yet still fostered great love and care.
Also, what's NOT in scope is same-sex marriage (by the way that doesn't necessarily mean that I endorse same-sex marriage) or even transgender people and gender identity (context is about reproductive actions). But more importantly, this is NOT an excuse for straight Christians to claim the "moral high ground" and dehumanize the LGBTQ+ community -- especially since there are a lot more sins that Paul will cover in the coming verses and chapters that ultimately declare everyone -- even straight Christians -- failing to meet God's standards in base form (which is basically the point).
I should probably revisit this once I do more research on the topic.