Learn Dutch: Het publiek juicht het hardst wanneer zij het record breekt. - The audience cheers the loudest when she breaks the record.

Because it describes how the audience cheers, making it an adverb. In Dutch, when you use a superlative to modify a verb, you always use the fixed word het followed by the superlative form ending in -st.

Does hardst mean "hardest" as in "most difficult"?

It can, but in Dutch, hard is also the standard word for volume ("loud") or speed ("fast"). In this context, het hardst means "the loudest". If you wanted to say "the most difficult", you would use het moeilijkst.

Why does the verb breekt go to the very end of the sentence?

The word wanneer (when) introduces a dependent clause. In Dutch, a dependent clause always pushes its verbs to the end. That is why it is wanneer zij het record breekt instead of wanneer zij breekt het record.

In English, "audience" can sometimes feel plural. Can I say het publiek juichen?

No. In Dutch, collective nouns like het publiek (the audience) or de familie (the family) are strictly grammatically singular. You must use the singular verb form, which is juicht.

Could I use als or toen instead of wanneer to mean "when"?