The phrase "fell over backwards" refers to putting forth an extraordinary effort to help someone or achieve a certain goal. There are several synonyms that can be used in place of this phrase, such as going the extra mile, bending over backwards, straining every nerve, and putting one's best foot forward. Other synonyms for this phrase include exerting oneself, doing everything possible, leaving no stone unturned, and going out of one's way. These synonyms all convey the same sentiment of a person putting forth their best efforts to achieve a goal or help another person, often to the point of sacrificing their own time or energy in the process.