You'll often encounter these sorts of broken lines in poetry or music because it helps to maintain a particular rhythm, pattern, or rhyme scheme. Consider students reading lines from Shakespeare, pausing unnaturally at the end of each line, vs an actor who reads through to the end of sentence so its meaning is clear.
In your example above, you'll notice the AA-BB rhyme scheme at the end of the lines, which is likely why they're wrapped as they are.
I'd agree w/Snailboat's interpretation as to where punctuation & obvious breaks would be if it were prose, but as stated you can also search for deeper meaning if you want. Playing with double meanings and alternate interpretations can be what makes it interesting.
On one hand, the singer is saying "Why me? I'm not the one she wants. I finally see my path to get clear of her and I can't stop now, so all she gets from me is a tear & nothing more."
On the other hand, even though he's "not the one," maybe he'd like to be and has "just begun" to care about her. Or he did before & is trying not to fall for her again. Maybe he's conflicted and torn in 2 directions by his feelings, just as the lines he speaks are torn down the middle.