Beginner Gardening: The Power of Peas, 0 by Pillita

Pillita wrote:
This summer, I had two flowerbeds sitting nearly empty and I got a wild hair to try peas. The ones I found were pink-eyed purple hulls. The peas produced were very good. My question is about the power of peas. I have read that peas have the ability to fix nitrogen and that they are good for cover crops, but I am a little unclear on what exactly that means for the soil and surrounding plants.

If I leave the pea plants in place (problematic to turn them under), will that benefit the plants that will be put in those beds next, even though they will likely be perennials, not vegetables?

Also, I'm curious about the cosmos shown in the picture. It is a volunteer amongst the peas. Nearly seven feet tall (DH is 6'2"), it is far above any other in the yard. Did the peas do this? Shall I have to turn into Pillita the Cowpea Fairy and merrily disperse them throughout all of the flowerbeds?

Lastly, I had someone tell me that the benefit of peas is to be found in the leaves. If that is the case, then should I just let the pea leaves fall as they may or will the value be lost once they fall off the plant? Thanks.