Why did the European powers and the USA intervene in the Boxer Rebellion?

There is, I believe, somewhat of a misunderstanding in your question, one that is unfortunately commonly held thanks to the colloquial use of 'Rebellion' to describe the Boxer crisis of 1899-1901. 'Rebellion' would imply that the Boxers sought to overturn the otherwise-established Qing government, and that the Europeans, USA, and also Japan interceded on the latter's behalf, but the political situation at the start of 1900 was considerably more complex. Complex enough that it's one of those rare occasions that it perhaps requires a diagram to map out. Now, I will note that this earlier answer of mine goes into more detail on the specific chronology, but as it addresses things from the perspective of the Qing court I'll devote a little more space here to explicating the foreign position.

To render the above into words, it is important to first note that we cannot divorce the Boxer crisis from the 1898 reform movement and coup, which saw somewhat of an alliance of convenience between conservatives and moderate reformers within the Qing government against the more radical reformers backed by the Guangxu Emperor, and an alliance with Cixi against the emperor himself. Elements of the army under Ronglu, possibly assisted by Yuan Shikai, launched a coup to essentially depose the emperor, and he would live out the rest of his life (until 1909, when he was probably poisoned) under house arrest. The imperial throne was effectively occupied by Dowager Empress Cixi, who was backed primarily by a mostly-Manchu conservative faction including her personal favourite, Zaiyi, styled Prince Duan. The more reform-minded moderate factions in the Qing government, consisting largely of Han Chinese officials in the provinces like Li Hongzhang, Zhang Zhidong, and Yuan Shikai, but also including some members of the inner court such as Ronglu, had backed Cixi as the lesser of two evils, but did not necessarily actively support her rule.

For their part, the foreign powers had not sat idly by in 1898. Arguably some of the impetus for the reform movement had been a reaction to the 'Scramble for Concessions' in the wake of the Juye Incident in December 1897: after the killing of some German Catholic missionaries by a group from the anti-Christian Big Sword Society, Germany demanded a lease on the city of Qingdao and its surrounding bay area, kicking off a series of lease claims by powers seeking both to expand their own influence and check each other's – Port Arthur (Russia), Weihaiwei (Britain), Guangzhouwan (France), and the New Territories of Hong Kong (Britain). Latterly, after the coup against the Guangxu Emperor happened, the foreign diplomatic community essentially refused to recognise Cixi's de facto leadership of the Qing Empire and continually protested her attempts to ensure a succession favourable to her own interests. There also remained a clear preferment towards the Guangxu Emperor, whose key supporters had found refuge overseas, principally in Japan. However, with no effective means of forcing a regime change other than military intervention, there remained an uneasy stalemate between various groups: the foreign powers and the Qing, and the imperial court and its erstwhile supporters.

The arrival of the Boxers would break this stalemate by forcing the various actors to pick sides. Boxer attacks on Chinese Christians and, more importantly for the Western powers, Christian missionaries, had led to the foreign powers adopting a somewhat paradoxical – some might even say hypocritical – stance. While they recognised that the Boxers were a domestic movement, the containment of which was ultimately up to the Qing, they did not actually recognise the Qing government whom they were insisting ought to deal with the problem. The moderate reformists saw the Boxers as a dangerous reactionary force that ought to be stamped out. But the Qing court ended up taking a slightly different view. The Boxers had declared their support for the Qing as a whole, which could, if the movement were officially co-opted, be specifically reframed as entailing support for Cixi's rule, and so the court was reluctant to employ force against them.

This in turn led to a vicious cycle of mutual paranoia. On the Qing end, for some time Cixi and her supporters had believed that the foreign powers had been angling at the restoration of the Guangxu Emperor to the throne, and were looking out for signs of a military buildup that might indicate that. On the foreign end, the lack of decisive Qing action against the Boxers in Zhili (while Yuan Shikai took quite harsh action in Shandong) suggested that the court was in some way actively colluding with the Boxers, and that attacks on Christians would potentially escalate to a broader attack on their presence in China – particularly their diplomats and their economic assets – leading to the foreign powers moving to escalate their military presence. This, of course, was taken by Cixi's faction as a sign of hostile intent by the foreign powers, leading to an increase in Qing military buildup against the foreign powers and reduced opposition to the Boxers, until eventually things tipped over when Vice-Admiral Seymour of the Royal Navy attempted to reinforce the Beijing legations via the Beijing-Tianjin railway, which was seen as the foreign attack actually happening. This in turn pushed the court over the fence towards an open alliance with the Boxers and a declaration of war against the foreign powers.

So it's not the case at all that the foreign powers intervened to protect the Qing from an uprising. Rather, they perceived that the Qing court – which they regarded as illegitimately usurped by Cixi – was colluding with the Boxers, and sought to expand their military presence to protect their citizens and assets. This became a self-fulfilling prophecy as the Qing court was thus no longer required to appease said foreign powers, and openly allied with the Boxers against them. However, the imperial court did so essentially unilaterally, and received no support from the non-anti-foreign, reform-minded moderate elements in the state, especially coastal provincial governors, who refused to be embroiled in the conflict and declared neutrality.