The 23-year-old, a former teammate of Oleksandr Zinchenko at Shakhtar Donetsk's academy, terminated his contract at Torpedo Moscow following the invasion of his homeland.
Kravchuk has already begun training with Man City's Elite Development Squad, with the move facilitated by fellow Ukrainian Zinchenko, and will stay with the Premier League champions until the end of the season.
"Leaving the club [Torpedo Moscow] was the only decision," the former Ukraine Under-21 international said.
"I felt really uncomfortable. People in Ukraine would not understand me if I continued to play there. I cannot believe it's happened. In the 21st century, in the middle of Europe. I still can't find the words."
Kravchuk recalled the time his mother called him whilst he was on pre-season with the Russian club in Turkey, after Ukraine first came under a series of attacks on February 24.
"Her first words to me were: 'Russia is bombing us'. It's shock. I was destroyed inside. I am in groups on my phone and always receiving messages with bombing alerts. Every time those alerts come through I am so anxious. Your only thought is that my family can die."
Kravchuk's family remain in the Ukrainian capital Kyiv, and the player revealed he spoke to his parents after being put through his paces on Thursday. His brother, Aleks, has joined the Ukraine's armed forces to fight Russia.
"I speak to my brother every day," he added. "I tell him every day how proud I am of him, for not only protecting our family but the whole country and the Ukrainian people. He is staying and fighting.
"My family are safe right now but my mother has just told me she can still hear the bombs. I'm really worried."
Kravchuk joined Torpedo Moscow last year after leaving Olimpik Donetsk.
Fellow countryman Zinchenko said: "While I wish the circumstances were very different, it has been good to be back with Andriy today.
"I know how much football means to him, and how it can help us during these difficult times. I would like to thank my club for giving him the opportunity to train with us."
Kravchuk is delighted at the opportunity to link up with Zinchenko in Manchester.
"I haven't seen Oleks for a while but he has always been helpful for me - even going back to when we were together at Shakhtar," Kravchuk said. "I'm really happy to be here with him."
Kravchuk's arrangement with Pep Guardiola's side was approved by the Home Office, Premier League and Football Association, with the player now searching for a new club ahead of the 2022/23 season.