For the dinosaurs in the audience, The typical use of this meme is when you'll be watching some sort of "you laugh you lose" or really any sort of commentary video/livestream where they might insert a funny gif or have a visual gag with some romantic or sexual edge to it, such as the one attached below.
Upon seeing this somebody will usually comment "hear me out" as if they are offering a defense as to why it's reasonable that they are sexually attracted to this character as part of an imaginary conversation that the reader would be having with them, where the reader is chastising them for having such thoughts. It's not that funny. It's just one of the many types of phrases that gets squawked by the "chat" in these videos.
I'm getting too fucking old, the fucking etymology on this is hurting my brain. I guess I grasp the concept, but wtf does a cake have to do with it. Why can't you just say you find someone oddly attractive or w/e.
The "cake" is a literal cake, and the concept is printed pictures on sticks or straws that you put into the cake as you mention or explain the unconventional person. It's just a "viral" online trend, a random fun thing friends do.
I know a woman that utterly hates that elf dentist guy. She's a lesbian though so maybe this lady is on to something. I need to understand the sexual energy at work here.