Glossary for "Yesterday" "Troubles": (noun) problems, difficulties "They're here to stay": They are [the troubles/difficulties] are not going to get better soon.
"Half the man I used to be": The singer regrets his words, and feels as though he is not who he was yesterday since he "said something wrong".
"Why she had to go?": May be ordinarily phrased, "Why did she have to go?" but was altered to fit the song.