In the given sentence, we are comparing the cheering of Kate's family with all of her fans. Since we are comparing more than two people, we need to use the superlative form. Therefore, the correct word to use in this case is "most loudly." The adverb "louder" would be used if we were comparing only two people.
To understand this concept better, you can refer to the provided grammar resources. You can visit websites such as grammar.ccc.commnet.edu, lousywriter.com, or uottawa.ca to learn more about degrees of adjectives and the comparisons of adverbs. These resources will give you a deeper understanding of when to use comparative (-er) or superlative (-est) forms.