In short, brown butter creates an intense nutty and sweet flavor that enhances just about anything you bake, sauté or you drizzle it over, especially chicken or seafood. It is a trick to get this right because there is a fine line between brown butter and BURNT butter, with the latter being disgusting. You might have a few misfires but once you master it this will become a staple of your cooking.
Take the amount of butter that you will need and put it in a sauté pan over medium-high heat and let melt, you will see it bubble and then brown. You should stir with a wooden spoon to scrape up the brown bits, then pour in a container and refrigerate. This entire process should take 6-7 minutes.