The two major notions are that Yuzu violates Nintendo's copyright [1] by allowing people to play unauthorized copies [2]. In order to do so it allows for bypassing Nintendo's encryption (by taking in the keys, it does not embed the keys in the software) it falls under a violation of the DMCA [3]. Essentially, trying to argue that the keys are copyrighted. Additionally, they claim that every user that has either dumped Nintendo games they lawfully owned and play in Yuzu, or have pirated the roms and played in Yuzu have violated copyright and thus Yuzu should pay up [4].
[1] "In effect, Yuzu turns general computing devices into tools for massive intellectual property infringement of Nintendo and others’ copyrighted work"
[2] "In other words, without Yuzu’s decryption of Nintendo’s encryption, unauthorized copies of games could not be played on PCs or Android devices. "
[3] "Recognizing the threats faced by copyright owners like Nintendo in the age of digital piracy, Congress enacted the Anti-Circumvention and Anti-Trafficking provisions of the Digital Millennium Copyright Act (“DMCA”), making it illegal to circumvent or traffic in devices that circumvent technological measures put into place by copyright owners to protect against unlawful access to and copying of copyrighted works."
[4] "On information and belief, Yuzu users have (1) dumped Nintendo games they have lawfully purchased and copied the game ROMs into Yuzu; and (2) obtained Nintendo games online from pirate websites and copied those game ROMs into Yuzu. Each such reproduction constitutes a violation of 17 U.S.C. § 501(a) for which Plaintiff is entitled to damages under 17 U.S.C. § 504 and injunctive relief under § 502"