Growing up, I heard "sit up straight" a million times, from my mom and my grandmother. Who knew, they were getting me ready for my most favoritest hobby later on in life!
According to this Runners world article I found, good posture plays a huge role in your running. When you run slouched over, you put too much weight on your lower body, mostly the knees. While this is done, your legs can't fully extend and your hip flexors contract. Also, while slouching, your diaphragm is contracted and you can't breath in as deep, which causes side stitches. I hate side stitches!
Check out this article, to see some workouts that will help improve your core, so you will not slouch!