How did Japan rapidly industrialize during the Meiji Restoration, and why was it allowed to do so by the at the time Imperial Powers like the UK, France, and the United States?

The first half of your question is already vast, and requires multiple answers.

/u/ReaperReader covered the big-picture economic history perspective in this answer: How/Why did Japan industrialise so fast and well?

/u/ParkSungJun discusses the more precise problem of financing industrialization projects here: Why was Japan able to industrialize so quickly and successfully hold off western powers when other powerful Asian countries like China, India and Persia failed?

/u/ParallelPain discussed the scientific foundations of the Meiji Restoration here: Japan managed to catch up very quickly in science and physics in the early 1900s, how did they accomplish this?

There have been a few other answers as well but I think those three are the best.

As for why the Western imperial powers "allowed" Japan to industrialize, they actually didn't. Starting in 1854, they imposed the same unequal treaties and treaty port system on Japan that China was getting. However, news of these unequal treaties caused outrage in Japan that led to civil unrest and swiftly, just 14 years later, a coup d'état. China eventually had a similar handover to the Republic of China in 1912, but this was a much harder process. Japan basically had some revolutionary ideas already beginning to catch fire in the early 19th century which made a complete clean-out of the old guard much easier in the 1860s. This did not in itself encourage the Europeans at all, but the new Meiji government was able to quickly encourage social, legal, and economic reforms with an eye to ending the unequal treaties.

Japan put significant resources into learning how to be taken seriously by Americans and Europeans to make technology transfer as swift as possible, and there were plenty of Western liberals willing to assist these efforts, despite Western tendencies towards cultural or racial chauvinism. As a result, by the end of the 19th century, Europeans were losing their ability to issue commands to Japan. Japan's final "unequal treaty" in a sense was the Triple Intervention of 1895 which tried to stop Japan from rising as a great power in the Pacific following its victory in the First Sino-Japanese War, but in 1902, Japan was able to obtain an alliance with the UK, and then won the Russo-Japanese War in 1905, which put an end to these attempts to keep it out of the imperialist great power system.