You're right, they should have explained it. However, I think it's the act of almost ending a loved one's life that made him truly tap into his "inner devil". If they had taken the time to explain that, even in a single line, it might have ruined the emotion of the moment. That's my theory, at least.
In the next game, full on DT should be present. It's obviously breaking out of Dante's body, and they probably have at least a framework of a story to build upon in the next game.
Edit: They can always explain it in a press release or something. The same way they explained the other aspects of the game.