The Eight Nation Alliance, composed of Italy, Japan, Great Britain, Russia, Germany, Austria-Hungary, and France, sought in essence to protect their interests in China and quell any attempt by the Boxers to take back control over the country. The Boxers rose primarily due to the fact that Qing China had failed to manage to keep Western influence 'out', and then proceeded to continue failing at modernisation. This, plus agitation brought forward in religious conflicts between Christian missionaries and government propaganda, as well as widespread drought and famine, essentially was brewing a massive storm in a bottle that was ready to burst at any second.

A backstory to this - on the 1st to 2nd of November, 1897 (about 2 years prior to the rebellion), armed bandits attacked three German missionaries and killed two, which prompted the Kaiser to send a force into Jiazhou Bay near Qingdao. This in turn caused several of the other countries who would later form the Eight Nation Alliance to also send their own forces into various parts of China and establish their own strongpoints. This answers your question on why so many countries got involved - they were all in on it together. By carving up China into so many small pieces for each country whilst having a quite obviously enraged populace, unless you all came together in support of one another, you would find that you'd soon lose what you came to hold.

The context of late 19th century China was of political and social turmoil. The Qing had already lost two wars to Great Britain and ceded many of its trade ports to the British, then the Taiping Rebellion, and then losing the First Sino-Japanese war, gave many the impression that China was becoming more and more foreign (bearing in mind that the Qing themselves, Manchu people, were already quite foreign to the majority of ethnic Han Chinese). And so, when Empress Dowager Cixi issued policies that supported the boxers, it indicated to everyone else her intentions, and in response the nations (following the defeat of the British intervention led by Seymour) formed the Eight Nation Alliance and together instrumented the defeat of the Qing dynasty of China.