It all comes down to fuel: New Horizons -- which is about the size of a baby grand piano -- had to carry over 3 billion miles worth of fuel to get into the same neighborhood as Pluto. That's already a lot of fuel, and fuel is heavy. The more fuel you carry, the more difficult and expensive it is to get your spacecraft off the Earth. Plus that extra weight slows you down, and nine years is already a long time to wait for data!
Why would we need extra fuel to swing into orbit? Pluto -- which is tiny, even smaller than our moon -- has really low gravity, which means it would be tough to get it to it to grab onto a spacecraft and pull it into orbit. And right now New Horizons isn't even on course to be in the right spot for Pluto to grab it.
To get New Horizons into orbit, the spacecraft would have to carry a second rocket that it could use to slow itself down and move it into Pluto's reach.
But it's okay: NASA expects to find out a ton of stuff during that brief flyby. Even if the mission only completes its top-tier objectives, we're going to get detailed maps of Pluto and its largest moon, figure out what they're made of, and get a good idea of what Pluto's atmosphere is made of, too.
And there's a silver lining to zipping on past Pluto. Pluto may be at the very outer edge of our familiar solar system, but it's on the inner edge of the Kuiper Belt, which is full of objects that remain mostly mysterious to us. With any luck, New Horizons will stay in good health and NASA will extend the mission, allowing scientists to observe other, less familiar objects even further away than Pluto is.
Updated: A previous version of this post stated that New Horizons traveled over 4 billion miles to Pluto, when in fact it traveled just over 3 billion.