TL;DR: Data rate is directly related to effective noise floor, which is related to bandwidth and received power (and lack of interference). Lower received power means increasing transmit power or reducing bandwidth (speed). APL had to choose the latter due to constraints. Read on for a much more detailed and long-winded explanation (sorry if too long winded!).
There are a bunch of inter-related reasons for this that boils down to Link Budget, which is basically how much signal you have above the noise floor to work with in your receiver. Thanks to the Inverse-Square Law mentioned above, the incredible distance to Pluto (at 32 Astronomical Units distance--1 AU is the Sun-Earth distance) means a very, very weak signal to work with regardless of transmit power.
New Horizons transmits TWELVE (12) MEASLY WATTS from each of its TWTA amps, which can be used primary-secondary or in a cross-polarized mode for 2 separate data streams. The full-size 2.1m/7ft high-gain dish antenna has a 42dBi gain, about the same as an SNG satellite truck's--only, instead of having to shoot to GEO at 35,000km or 22k miles, you're trying to shoot SOOOOOOO much further at around a tenth (-10dB) of the transmit power. On the plus side, they will be using the largest Deep Space Network dishes to receive and talk to it, which are 70 meters in diameter, but that only makes up so much for all this.
The lack of power is due to using an RTG (Radioisotope Thermal Generator) which uses the heat from decaying Plutonium-238 (one neutron less than the Pu-239 used in nuclear fission) to supply a thermocouple setup. The documents state just under 4kW thermal output and around 245 watts electric at launch, dropping to 200-220 watts at the Pluto encounter. That little bit of electricity has to be able to supply EVERYTHING on the spacecraft, and it decreases with time, which is why the Voyagers can no longer send images or much else that requires significant power; NH will end up the same way eventually.
To come full circle, the massive distance and low power means that there won't be much signal left by the time it arrives at either destination. The effective noise floor is directly affected by the bandwidth of the signal, which is why automated transmission systems--like the modems in your cell phones, land lines, etc--have both fast rates for good-quality links and slower rates for weaker links: this is especially true with LTE. Ham radio operators (which I'm one) have use CW and Morse Code for talking using moon or aurora bounce because the distortion is too bad to use any kind of voice mode, and now there are new data modes that optimize for VERY SLOOOOOOW data rates (one bit per second or less) that exceed CW-by-ear for reliability by sacrificing bandwidth and data rate, but this obviously would be useless for this type of science mission other than health telemetry.