If you define worry as "the state of being anxious and troubled over actual or potential problems", then I'd argue that worrying in itself has no positive effect on the 'real' problem anyways.

The only mechanism by which 'worry' might be useful is if it triggers the problem-solving part of the mind and ultimately leads to actions that solve this problem. In my opinion this mechanism is highly inefficient and in fact often backfires: we focus on the worry instead of thinking about solutions, let alone implementing these solutions.

Meditation, and a mindful approach to life is first of all useful as a debugging mechanism. As you train your ability to direct your thoughts and pause them, you learn to discern what kind of thinking is useful, and what isn't.

So, for example, you might realize that you've been caught up in worry and just the act of being aware of this pointless train of thought might prompt you to sit down and create an action list of how to get out of this situation.

Another benefit is that frequent meditation somehow improves the connection between the thinking mind and the subconscious, as well as the body. I've had moments where meditation helped me suddenly realize a solution to a problem that I couldn't previously solve. And almost every time that I meditate, I will be aware of hunger, or thirst, or some other feeling or physical need that I would otherwise completely neglect.

Now I'm a particular type of person, so these benefits might be more explicit to me, but from what I understand most people will experience these things to some extent.

I don't have anything to say, other than I really enjoyed your reply and think the debugging analogy is great!

There's no one answer to that, because "how" is dependent on the individual and their relation to their thoughts (i.e. my thoughts are probably organized very differently from yours).

Furthermore, it is hard to claim that one would be able to "alter their thoughts" as directly as your sentence suggests, but meditation at least allows the recognition of separation between the "worry" in the mind and the "problem" in reality.

For some people, this recognition leads to a sense of control over that worry, which might be one example of your "how".