What role does editing play in the writing process, and how can writers develop strong editing skills?

Editing plays a major role. Editing is where your work comes alive. I said this in an earlier question, but it's better to just get something on your page. The bulk of your time should be spent re-reading what you've written and refining it. 

This is where a good plan (outline for nonfiction and plot for fiction) comes in handy. If you know where you're going, it's easy to tie up the loose ends when you get there. You just have to put your head down and write. 

And limit distractions! I set timers (this is along the lines of the Pomodoro technique) so that I can focus on what I'm doing. The Wi-Fi on my phone gets shut off during these intervals and I am 100% focused on editing my writing. When the timer rings, I come up for air, check my Fiverr messages, and get stuck back in.