What does ‘Train-wreck style’ mean?

Solution 1:

The idea of using train-wreck to describe a person comes from not only the awful calamity of a literal train wreck, but also the macabre pleasure that the general public seems to derive from viewing such a spectacle. It is the morbid interest that we have in watching other people destroy their own lives that really gives "train-wreck" its meaning.

"Train-wreck style" is an extension of this idea. In your example, Anne Hathaway's character "offers an indelible, if sometimes repellent portrait of a recovering addict who makes people squirm." Her character is the train-wreck; the speech she gives at her sister's wedding is delivered in a "train-wreck style" because we can all see the awkward and embarrassing end of her disastrous attempts to heal from her addiction, but we can't help but watch her anyway.

EDIT:

You asked whether "self-destructive style" could be substituted for "train-wreck style" and it is similar, except that "train-wreck style" is publicly self-destructive. You cannot be a train-wreck in private, although you might be very self-destructive in private.

Solution 2:

It's not so much the style or character of the individual, but the nature of the wreck. A train wreck is usually something you can watch happen for a long time, helpless to stop it. Trains don't stop quickly, and they're on rails, so they're not free to steer around the problem. There is a point, a long time before the actual wreck occurs, that you know it's already happened, so to speak -- all that's left is to watch the physical manifestation of a probability wave that has already collapsed.

In the case of habitual, overt self-destructive behaviour (as demonstrated by the late Ms. Winehouse) it is, perhaps, easier to see how the metaphor fits. In Ms. Hathaway's case, it was only her public image that was destroyed, but the commenter was saying that once she began speaking she had already committed too much, there was nothing she could do to ward off the disaster.

Solution 3:

It means, approximately, "in a manner tending towards disaster" or "in a way which can end only badly."