"I agree with the decision, but I came to my decision via different logic."

He thinks that the majority reached the right conclusion (the result - that the defendants were entitled to the money) but not for the right reason (the particular argument that the majority opinion made)

It means he agrees that the money should be returned, but not for the same reasons that the other justices gave. So he'll have his own section in the judgement explaining his own reasoning.

SCOTUS rulings are not yes-no votes. There are any number of opinions written and justices who took part in the case (usually all of them) will sign on to one of these opinions, while sometimes noting that they largely agree with another one. I didn't look up this particular case but usually that sentence means that there were at least two opinions written: one agreed by a plurality, one by Alito, and zero or more additional opinions (at least one if there were any dissents). It means that Alito's opinion amounted to the same yes-no answer(s) as the majority but he was unwilling to say he agreed with their actual written opinion.