Japanese started to eat Beef after 明治維新 (Meiji Restoration) started. The foreign ships came to Yokohama and brought a new style of eating. At that time, cow were work force protected by Buddhism, so no one eats cow.
1872/1/24, Emperor eat beef for the first time. Was a huge news during that time and eating beef is a sign of new civilization. The beef hotpot (牛鍋) made of simple ingredients with Beef, soy sauce, and sugar. In just 5 years, there are 550 shops who sell 牛鍋.
The name changed after 関東大地震 (1923 Great Kantō earthquake), everything was destroyed and people have to grill and eat things on 鋤 (spade), hence the name すき焼き1。 People moved around in Kansai and Kantō after the earthquake, and the Beef hotpot got cross influenced.
The slicer was not invented yet in the time. So 牛鍋 are consists of beef cubes instead of thin slice beef. Where you can still eat the original 牛鍋: