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Hopefully this doesn't confuse you more. It is important to remember that Japanese only has 2 verb tenses(past and present). In order to say something in the future, they would use details like ashita(tomorrow). The question then is "how do you say something in the present that describes a future event or process?". This is known as a present participle. "I am typing this message." In English we take the verb and add -ing generally. It gets complicated if the verb is passive(to be). We don't say "I am aming a doctor." Rather "I am becoming a doctor" So you can think of (koto ni) or (you ni) as -ing ending to become. So literally translating Nihon ni iku koto ni natta would be
"To Japan to go becoming". Obviously we wouldn't say that in English but the meaning is there. Subjects are implied a lot in Japanese, in this case it would be I (watashi wa)
So you'd get "It is the case(it's been decided) that I'm going to Japan." Since starting a sentence with an indefinite pronoun is considered bad English "I am in a process of going to Japan." With "Undou suru you ni natta"
"Excercise to do (exercising) becoming" would be "I am in the process of exercising." "I started to exercise" has a similar meaning. Correct me if I'm wrong but that's what it sounds like.