Long distance running requires a lot of stamina and mental strength. Some people say it is relaxing.
How can it be relaxing? 
Well, humans are well adapted for running significant distances so I think once someone does it for a while they get use to it and it becomes something they like to do.
How does it help one mentally? I could use some help, I have mommy brain. 
I'm not sure I couldn't find that but, I do know that Long-distance running is only considered long distance if you run at least over 5 miles
How do you get started with long distance running? 
You have to pace yourself. First start out with smaller distances like 1 mile, 2 miles, 3, etc. Eventually you'll get to continuously running over 5 miles and once you do that it will be easy.
Do you need special shoes for running? 
Running shoes are definitely a plus. For instance you wouldn't want to run 5 miles wearing basic sandals that have a thin sole and a simple strap. You need some nice running shoes so that your feet won't hurt.
I wear flip flops year round so would need to invest in a good pair. Anything else I would need?