When people use that word, they are trying to convey to others they're being earnest, because it's the opposite of sarcasm, which they convey as irony.
Irony has a loose definition and can mean many things depending on the context, such as situational irony, but whenever anybody uses the word "un-ir0nically", you know that they mean they're being sincere, genuine, honest, earnest, etc. There are other words to describe their authentic thoughts than using that fad of a word. It has a more strict, rigid definition than the word 'irony'.