One of the tricks is that sometimes when you hear a song in your head, it's a bit sloppy and has some blanks in it. Like, there may be a melody with a descending run, but it's not clear what the steps are in the descending run or which notes it starts and ends on.
Another one of the tricks is that people typically focus on the topline melody, but if you hear a song in your head, you probably also at least "hear" the bass, or even other internal voices in the harmony. But without practice, you might not know that this is what you're hearing in your head, so you might struggle to put it to paper.
For a basic example from the Beatles, take the first line of the chorus for Yesterday, "Why she had to go". The melody holds the same pitch for the first two notes "why" and "she", but the bass is descending. If you know the song Yesterday, you can probably hear that descending bass in your head. If you can translate that to paper, you can transcribe songs. If you can hear songs in your head and transcribe them, you can write songs.