Can 이별하다 mean "to be hurt"?
The sentence is 왜 혼자 사랑하고 혼자서만 이별해. I understood it as "Why am I loving alone while you left alone". However the translation on the site says "Why am I in love alone, why am I hurting alone". Doesn't 이별하다 mean "to part/to separate"? Can someone explain this sentence for me please? Thanks! :)

Question

Answer

The direct translation would be : "Why do you fall in love alone and then break up alone"

Answer

'I'm hurting alone' is not correct. 
You are right. 이별하다 means 'to part/to seperate/to break up'

By the way, "Why am I loving alone  you left alone".  is slightly incorrect.
The sentence dosen't mean  but just .
> Why I love alone, why I have to say good-bye alone(/why I seperate alone?)

We can't know the exact intention and reason cause it's a kind of metaphorical(or emotional)  expression though, we can guess that he is in a onsided love or she just went out(played) with him without taking him seriously.