I hope they change the English sentence soon because it's currently nonsensical due to a missing word. I thought it meant "I was left behind to write a letter" or "I was left there to write a letter".
As you say, this is not the meaning of the French sentence. "All I had left was to write a letter" is good as well, but not as clear without "to do".
Because here il doesn't refer to a person. It's like the construction Il pleut = It rains. We call it Pronom impersonnel (Impersonal pronoun). The expression Il reste (quelque chose) = There is (something) left, whereas Il me/te/lui... reste (quelque chose) = I/You/He/She... is left (with something/to do something)