Just to put some perspective on things:
1. New Horizons is really far away from the Earth.
At the moment of closest approach, New Horizons was over 6,600,000,000 kilometers away from Earth. This is about 6 light-hours. And the spacecraft is continuing to get farther by about 14 kilometres per second.
2. Transmissions from farther away are weaker.
The inverse square law states that the intensity of things like radio signals and sources of light (energy per unit of area perpendicular to the source) is inversely proportional to the square of the distance. That means doubling the distance results in us receiving only a quarter of the energy.
3. New Horizons only has so much power to work with.
The spacecraft is powered by a single RTG (radioisotope thermoelectric generator) that contains ~11 kg of Plutonium-238. At launch, this produced 245 watts (at 30 volts of direct current) of power, but due to radioactive decay, this decreased to 200 watts by the time of the July 2015 Pluto flyby, and further to 190 watts by the time of the January 2019 MU69 flyby.
For data transmission, it has a 2.1-meter diameter high gain dish antenna, a 30-centimeter diameter medium-gain dish antenna, and two broad-beam, low-gain antennas. The high-gain beam is 0.3 degrees wide, and the medium-gain beam is 4 degrees wide (used in situations when the pointing might not be as accurate). New Horizon's radio system is powered by a TWTA (Traveling Wave Tube Amplifier), which consumes 12 watts. (That's about the same as a modern CFL light bulb!)
There are actually two TWTAs for redundancy; one with left-hand circular polarization, and one with right-hand circular polarization. After launch, they figured out a trick to use both TWTAs at the same time, which increased the data transfer rate by 1.9 times. They used this two-TWTA mode to get all the data back from the Pluto flyby more quickly.
4. There's a limit how sensitive the antennas on Earth can be.
Even though we listen for New Horizon's transmissions using enormous 70-meter dish antennas from the Deep Space Network, there comes a point where it starts getting difficult to discern the signal amongst a sea of white noise and other interference, because the signal is so weak.
Here's the 70-meter dish from Madrid. It's hard to do much better than this.
5. So, the downlink speed has to be restricted because of the very weak signal.
As elaborated upon in The_Sympathizer's answer, the signal-to-noise ratio gets lower when the signal gets fainter, and so you have to transmit data more slowly in order to make sure that the data you receive is correct.
NASA has a neat interactive page that shows what each antenna in the DSN is doing right now. Here's a screenshot from January 3, 2019, 01:11 UTC:
As you can see, the signal that this dish is receiving from New Horizons is only 1.29E-18 W in strength. That's 1.29 attowatts. That's extremely weak.
So, as a result of the faint signal, it looks like the people at NASA decided to restrict the downlink rate at about 1000 bits per second (125 bytes per second), as an optimal balance between data integrity and downlink speed.
As a point of comparison, the https://google.ca homepage (when you're not logged in) comes out to about 1 MB. So, if you tried to open the Google homepage at the speed of the New Horizons downlink, it'd take over 2 hours for the page to fully load.
6. There is a lot of data.
New Horizons was busy during the flyby. It collected about 50 gigabits of data (6 GB). So at 1,000 bits per second, on-and-off (the solar conjunction that Luis G. pointed out will also briefly delay the data transfer), it'll take about 20 months for the full set of the Ultima flyby data to be sent back to Earth.
- During the Pluto flyby in July 2015, downlink speed was at about 2,000 bits per second, and it took about 15 months to download all 55 gigabits (7 GB) of Pluto data.
- During the Jupiter flyby in February 2007, downlink speed was at about 38,000 bits per second.
Further reading: Here's an interesting related question: How to calculate data rate of Voyager 1?