What we're looking at here wasn't originally produced by a Qing subject, but rather a British one living in Hong Kong – look closely and you will see that there is a lot of English, albeit of varying quality. This more famous version of the cartoon is a later version of the 1898 original, which can be seen here, which includes more Chinese text and more accurate English.

With that out of the way, what do the the various animals represent? The simple answer is the 'spheres of influence' of various foreign powers – we see, for example, the Russian bear, the British bulldog, American eagle and, amusingly, the French frog (which further illustrates the degree of British influence involved). The US had seized the Philippines in the Spanish-American War, France held Indochina, Britain held Hong Kong, and the Russians were, of course, running the Trans-Siberian Railway through Manchuria and maintaining substantial armed forces there. Japan's comment that 'John Bull and I will watch the bear' is a reference to the Anglo-Japanese Alliance, intended to keep Russia at bay. The 'rays' of the Rising Sun which quite distinctly fly over Korea are in reference to it having been overrun (but barred from actual conquest by Russia) by Japan in the Sino-Japanese War of 1894-5, whilst the cable linking it to Taiwan references the actual seizure of the island. In the bottom of the picture, a hare representing Germany can be seen waving the German flag, which also flies over the German concession at Jiaozhou, encircled by – unsurprisingly – a sausage.

What of the three men in Xinjiang and Tibet? Well, it is first important to remember that the Manchu Qing were not Han Chinese, but foreign as well. To quote Julia Lovell's The Opium War (2011):

The story of the Qing is of a great colonial enterprise, in which a Manchurian conquest minority somehow kept in check for over two and a half centuries a great patchwork of other ethnic groups: Chinese, Mongolians, Tibetans.

What is notably missing is, apparently, a key which explains the various sections. Not being aware of what this key says, I can only refer you to this 2011 English-language article by Rudolf G. Wagner in the German journal Transcultural Studies, which is, thanks to access to the key, able to go into more detail. In short, the men in Xinjiang only make sense if you include the idle Mandarin in the centre, trapping the two men in his net, restricting them to a military (left, represented by the bare chest, horse and weapons) or civil (glasses, book and chair) career. There are numerous other aspects, of course, like the antiquated nature of the weapons as a commentary on Chinese military capability, the banality of the characters reflecting on the difficulty of learning Chinese – especially classical Chinese – as a second language, etc., but there's relatively little I can add to Wagner in terms of straight-up decoding the image.

To some extent, the history of the Qing dynasty is one, in large part, of race relations – to quote Lovell again:

Chinese-language sources accounts of the Opium War reveal a divided, distrustful society, with practically very grouping in conflict with another: Han Chinese with Manchu officials; northern Chinese with southern Chinese; central southern Chinese with deep southern Chinese; provincial gentry with the central government; and the increasingly desperate hoi polloi with almost every group they encountered.

Anti-Manchu sentiment would rear its ugly head again not just in the Taiping Rebellion, where racist language permeated throughout Taiping publications, but also in 1911, which was rife with massacres of Manchu civilians. The updated cartoon here is not merely in opposition to China's new foreign devils, but also to her old ones – those sitting on the Dragon Throne.

Aside from direct references to Lovell what I've written is a bit of a hodgepodge based on various bits of reading absorbed over the years, but many of these are good places to look for an understanding of the wider context.

• Julia Lovell's The Opium War (2011), whilst centred on its eponymous conflict, is actually a pretty good introduction to both Qing rule in the 19th century and how the conflict was framed by successive governments.
• Robert Bickers' The Scramble for China: Foreign Devils in the Qing Empire, 1832-1914 (2012) is slightly underwhelming in that the focus is largely on the British presence in Shanghai and Hong Kong, but might serve as a useful basic framework.
• Tonio Andrade's The Gunpowder Age (2016) includes an assessment of Chinese military developments (and lack thereof) during the 19th Century, whilst Kwong Chi Man and Tsoi Liu Yun's Eastern Fortress: A Military History of Hong Kong (2014) is quite an enlightening look at British colonial defence policy.
• Other books that were of either direct or indirect relevance to my answer were Richard Connaughton's Rising Sun and Tumbling Bear (2003) on the Russo-Japanese War and Stephen R. Platt's Autumn in the Heavenly Kingdom (2012), particularly its sections on Han-Manchu enmity.

This image most likely expresses the "open door policy" that the major Western powers had towards Qing China during the age of New Imperialism in which nations would attempt to establish "spheres of influence" in China where they would indirectly control trade and the political balance in these regions, but not actually own any of the land. I am not sure however, what the three figures on the left represent. Perhaps the inability of the Chinese gov't to deal with Western influence?