I do not see how this question relates to History subject, but anyway I will answer it because you just could have misplaced it and i could still be useful for you in the English subject.
First let us start by explaining why the other words cannot complete the sentence:
The word loosely is used mainly to talk about things that can or cannot fit into a place, for example clothing, they can fit loosely.
Happily could be used as a literary device for personification to give the water the ability to "feel" happy with the meaning of flowing without problems, but since there is no justification for it in the sentence, we cannot use the word.
Densely cannot be used because it would mean the opposite to the normal reaction caused by heavy rain, when a liquid is dense it moves more slowly.
So the only possible and correct answer is:
1.- swiftly
the heavy rain caused the waters in the little creek besides the park to flow much more swiftly than normal.
