Here are a few examples of how running other games helped me improve my DMing:
1. Early on in my gaming career, I fell in love with Rolemaster, and all its complex tables and crit charts. But some of my friends were not so enamored, no matter how much I poured into GMing the game. What I realized was that crit/fumble charts are more fun to read than actually play with. And most importantly - it dragged the game to a halt. Meaning, combat slowed and that made the players unhappy. When we moved back to D&D (2e), I did everything I could to streamline the combat - and the players loved it. So I learned a lesson for my table at the time, streamlined combat that doesn't take 2 hours is the way to go. If we would have only played D&D, I doubt very seriously I would have ever tinkered with the system to make it more streamlined, and in turn, bettr for my players.
2. When I was younger, I had a preference - Middle Earth style play. I couldn't stand when players were suddenly cats or orcs or lizard men. Then I bought Gamma World and Earthdawn, learned the rules, and we played those for awhile. Earthdawn had windlings (pixies) and some reptilian creature. Gamma World had mutated animals. As a player in Earthdawn , I grew to really like the pixie and my reptilian friend. As a GM in Gamma World, I didn't mind the mutated animals running around with humans. Then we shifted back to D&D. I truly saw the game in a different light, and that allowed my players to express who they wanted to play more freely. Hence, I became a better DM for my table because of it. (It also allowed me to accept settings like Expedition to the Barrier Peaks, as I was opposed to sci-fi mixing with my fantasy back then.)
3. Rules, rules, rules. To be a good DM, you need to know the rules. It allows for fair calls, and even more important, consistent calls. Some games are unplayable if you do not adhere to the rules. The ones that come to my mind are Pathfinder (1st edition), Vampire the Masquerade, and Warhammer. Learning these allowed me to see the breaking point in games much more easily, and therefore, avoid them during play. It allowed me to see bending points I never would have noticed had I not been exposed to those other rulesets. And that in turn allowed me to apply the rules in a more fair and consistent manner.
4. Creativity. This one may not be as true today as it was 20 years ago, but it was definitely a learning experience for me. I remember thinking I was being creative and then picking up Numenera. It opened up a whole new world of magic items for me, and many of those magic items my players loved. We were playing 4th edition at the time, and I never would have thought of some of those magic items. It made our game more enjoyable.
Sorry for the rant, and I am almost done. But please allow me to counter before the obvious rebuttal comes. I know anyone can say: "I already let all races play, and already have short combats, and know the D&D rules and apply then fairly, and am already super-creative." But the point of those examples isn't that you would learn the exact same things as me. The point of the examples is that exposure to alternate systems expanded my repertories. It enabled me to learn DMing skills I would not have been able to had I just stuck with D&D. The point is, you don't know what you will learn until you try.