The horizon is Earth's horizon ... absent any obstructions such as buildings, trees, etc. Or another way to think of it is the horizon you would observe looking out across the ocean while standing on a beach.
The notion that it is "10° above the horizon" means that from the horizon line, it would be 10° up. You closed fist (as if you are about to punch someone) measure about 10° (from your thumb to your pinky finger) if held at arm's-length. Close your fist and extend your arm ... placing the bottom (the edge along the side with your pinky finger) on the horizon. 10° up is roughly where the knuckle of your thumb is located.
Caveats: Time and Date
But a couple of cautions should be mentioned regarding the article...
Objects in our Solar System move from night to night and this motion can sometimes be noticed even from hour to hour. Tonight (July 21, 2020) it is still below the "bowl" section of the Big Dipper asterism. But in a few days that will no longer be true. One week from today it will no longer appear "below" any part of the Big Dipper asterism. It is heading through the Coma Berenices constellation.
Also, the 10° measurement... this assumes both a time in the evening AND a latitude on Earth.
I'm located at roughly 42° North latitude. At 10:30pm for me and tonight (because time and date both matter) the comet is 22° above the horizon. But if I wait an hour and check 11:30pm ... comet is only 14° above the horizon. This is because the comet is setting ... sort of. Technically if you are far enough North the comet doesn't "set" below the horizon. Since I am located at roughly 42° North, anything with 42° of the celestial pole (near the North Star) doesn't actually "set" below the horizon. It will be up all day and all night. But as dawn approaches and the twilight brightens, the sky glow will wash out the comet until it is no longer possible to view from any "daytime" part of the planet.
Rather than trying to imagine where the comet is, there's a much easier way to just nail it.
Technology has come a LONG way over the years. These days, you can get an app for your smartphone that will tie into the GPS, compass, and level/tilt/gyro sensors on your phone. It's amazing how well this stuff works. You hold the phone up to the sky, and it knows everything in the sky that should be visible from your current location and in that direction. It will literally draw the sky and show you the precise location of the comet.