Why does a person's mind keep imagining situations with problems that do not exist and continue solving those problems in the imagination?

It sounds like you're talking about what psychologists and neuroscientists commonly call episodic future thinking or mental simulation. In this cognitive state, we simulate possible future scenarios, often pertaining to social interactions we might have, or survival situations. This allows us to pseudo-test hypotheses regarding the possible outcomes for an imagined scenario based on inference and the memories of our past experiences in similar situations. This may be adaptive if it allows us to prepare contingency plans for those scenarios, should one arise. We do this habitually, perhaps because there are so many possible scenarios to simulate, and the more we explore options and form tentative contingency plans, the better prepared we are for the world. It might also just be a rewarding activity in and of itself.

Absent-mindedness, mind wandering, or distractibility by one's own thoughts may be problematic if we are chronically distracted away from the present moment and the tasks that need to be accomplished at the moment. Mindfulness therapies may be helpful in training the ability not to let the mind wander and instead focus on desired tasks. Activities that require concentration may also train this ability, such as music performance. In severe cases, stimulant medication may be helpful, such as amphetamines or methylphenidate.

Regarding neural mechanisms for these states, there has been quite a lot of research and speculation in recent years on the interplay between two major brain networks, typically called the default mode network and the salience network. In functional connectivity MRI studies, the default mode network seems to be active during the state you described, whereas the salience network is active during focused attention on the external world and while completing a task. It may be that differences in how these networks have developed in an individual dictate the strength of their tendency to internally simulate rather than focus on the external world.

Here's a study regarding these networks and PTSD: Neural Dysregulation in Posttraumatic Stress Disorder: Evidence for Disrupted Equilibrium between Salience and Default Mode Brain Networks

There is something I heard of can be done but don't know how effective it is(it seems like it could be effective but I haven't seen any actual research done on it, just a bunch of theory that's honestly a little bit woo woo). In a book called "The Inside Out Revolution" by Michael Neill he talks about the idea that all of our experiences are our thoughts/neural impulses in the brain and the attempt to correct/to fix the thoughts is the thing that drives us down a bad spiral.

Imagine if you were to sit in a blank room for 15 minutes, no artwork on the wall, no music, no phone or anything else to keep you distracted. If you are the average person, during this time you would have experienced many different things(pizza sounds good right now/ do they hate me/ I wonder if peanut butter has real pennants in it/ etc). Even though nothing in your surroundings was "causing" you to experience any of these thoughts/sensations. They came about as a result of your thoughts/neural impulses/imagination and faded back away into nothing.

The idea he proposes in this book is a realization that everything we experience is a function of our thoughts first and not our environment, when you start having this insight the terrifying/nagging/constant/repetitive negative thoughts carry much less weight.

I mentioned earlier that the thoughts come and fade back into nothing, his theory is that some disorders are caused by this broken loop. If a person takes some of these thoughts too seriously they will fixate on them more and more and not allow them to fade back into nothing as they normally would as do other random thoughts/neural impulses. He makes the argument that a lot of disorders are caused by a correction mechanism gone wrong, instead of having the insight that the thoughts we are experiencing are just that... thoughts, we take them seriously, we think that the thoughts are a function of our environment/circumstances because we fail to realize that the causal mechanism for our sensations are mostly our own random neural impulses.

According to the author, the reason why people keep on trying to solve their problems in their minds is because the thoughts we all have every day feel like they are real and we haven't learned to recognize them as just our thoughts