This phrase is not correct, because it means that the one who says it is putting his trust in himself first. What he should say is, “I have done my best and I ask Allah for help.” This is what is correct.
Perhaps when a person says “I have done my best and the rest is up to Allah”, what he means is what I have suggested here, i.e., what I could do, I have done, and what I cannot do is up to Allah, But the phrase is basically incorrect. Rather he should say, “I have done my best and I ask Allah for help.”
And Allah knows best.