Why would I want an external recorder/monitor?
If you've just started out in video, on-camera monitors may seem like a luxury. Experienced videographers will tell you otherwise. As well as enlarging the display to make recording easier, lots of on-camera monitors can also improve your camera's video output. Whether you're shooting on a cine camera, a mirrorless or a DSLR, an external monitor is a necessity if you want to get serious about filmmaking.
Why would I want an external monitor/recorder?
There are two main benefits to using an external recorder: to get a bigger, more informative preview as you shoot and to capture better quality footage.
The monitor side of things, there are a lot more benefits than just having a bigger screen to see things with, though this in itself is valuable. The ability to see your scene on a larger screen makes it easier to spot small, distracting objects and check precisely where your focus is set. It can also help you better visualize the way your final footage will look, helping you make creative decisions such as choosing how much depth-of-field you want.
It's also common for monitors to offer overlays and composition aids. For example, framing guides that show crops for different aspect ratios can be helpful if you intend to publish your work in something other than the camera's native aspect ratio.
Also, freed from having to share battery power with so many other functions, external monitors can often be run brighter than the rear screen on your camera, making it easier to shoot outdoors.