You won't find 'kładać' in a dictionary. There is 'kłaść' which is to put down, e.g. 'Kładę ubrania na stole' - 'I'm putting the clothes on the table'. 'Wkładać' is to put inside something - here, inside the box.
How would you say "to put something", as in my example: "I am putting the flour into the cabinet?"
"Wkładam mąkę do szafki." Mind what was discussed above: "wkładam" into something, inside; "kładę" on something.
Hm it's not that bad but strange. It somehow suggests me that this „box” is not stiff, that you can wrap it around the „clothes”.
You can't put something into without putting it in, either should work.
Reported error as hints include plural, yet marked as wrong. "~clothes into boxes."
Hints 'include' plural, as you wrote. "pudełka" is the Nominative (and Accusative) form of "boxes", true. And also the Genitive singular. This happens for most neuter nouns, actually.
And as the hints are for the whole course, and not specific sentence, they are all displayed here.
"do" takes Genitive, so the only possible interpretation is a singular box.