... "I'm DAREDEVIL"? I mean, what was the point of showing her the mask? Didn't he think that KAREN would understand what it meant?
I know it may seem like a detail, but to me it ruined the emotional impact. Not to mention that it felt like an IRON-MAN rip-off.
No one knows what super sense are or that Matt has them.
Karen simply wouldn't believe Matt is daredevil. At first.
An adorable blind nerd telling her he's an ubermensch vigilante. Yeah right.
That's not even possible.
Foggy has been telling Karen Matt has a drinking problem and is falling down injuring himself.
I just wrote a pretty long-term analysis of this particular moment, though it's more about Karen's end of the interaction.
http://www.imdb.com/title/tt3322312/board/thread/263753335
As for why Matt would admit his secret to Karen? well, his personal life is at rock bottom. Stick left, Elektra's dead (that he knows of), Frank Castle is in the wind, Foggy's leaving for another firm. At this point, his time with Elektra has practically alienated him from his two closest friends. He probably feels that he needs to take the first step to rebuild his life. He's realized that keeping secrets from his friends actually makes things worse. Throughout season 2, you have to remember that Foggy knows about Matt's double-life as Daredevil, but Karen doesn't. Keeping Karen in the dark about all of his strange activities, like what Elektra was doing in his bed, or his strange absences and frequent injuries. He had nowhere to go but up from his point of view. Either she says "yay" and they start going out again, or she runs from the situation and he is where he is now anyway.