It's partly the same issues as the launch problem. If you put more fuel in the fuel tanks of the rockets, you then increase the mass. Then to lift that fuel you need to add a bit more fuel to lift that fuel, and so on and so forth.
A similar problem exists with the current propulsion system on probes but before I go into that I'm going to (very briefly) explain travel in space so we can understand the problem.
Travel In Space (assuming traveling in a straight line)
Travel in space is not the same as travel on land. Travel on land requires the constant burning of fuel to be able to replace the speed lost to friction, air resistance, etc.
Travel in space doesn't work the same way, it doesn't require a constant burning, it requires you burn enough fuel to propel the mass to that initial speed, then enough to burn in reverse to slow itself down at its destination.
(Which isn't half and half, it requires more burned fuel to speed up than to slow down, as the slowing down portion has less mass because we can discount the lost fuel burned to accelerate in the first place.)
Back to the problem
Ok so we could just add more fuel to the probes, but then we run into the same issue as the launch issue, adding extra fuel adds extra mass which requires extra fuel of its own to burn in order to propel the increased mass to the same speeds that we wanted.
So really what we want is a method of propulsion other than the existing liquid and solid process.
As you can see here, NASA already has some alternative ideas they want to try out for propulsion, I will briefly cover them below in case the link dies.
Nuclear Thermal Propulsion
Nuclear thermal propulsion – heats a fluid, usually hydrogen, in a
high temperature nuclear reactor that creates thrust to move the
rocket in space
NASA expects this type of propulsion system to be much lighter and a more efficient method of propelling ships in space.
However, every silver lining has a cloud and this is no exception. What currently stands in our way of using this system is the extreme difficulty to keep hydrogen in it's liquid form.
As you can see here, hydrogen needs to be kept at 20 Kelvin to stay in a liquid form. This proposes many technical challenges, first to reduce the temperature to such a level, then again when trying to stop the liquid fuel from heating up from the high temperatures of the exhaust!
And don't be mistaken, the technical problems with using liquid hydrogen isn't for lack of trying. in fact the idea to use liquid hydrogen as a fuel has been around since at least the 1950's!
Plasma based propulsion
NASA is also investigating a plasma based propulsion system called project VASIMR.
The idea is to use a nuclear reactor (again) and hydrogen (again) to ionize the hydrogen and blast it through a magnetic nozzle.
Obviously this is very technically challenging but there is also the issue that the plasma has to be magnetically shielded form the hardware of the ship or it causes electrode erosion in the engines themselves.
(Excuse my lack of knowledge of the physics around how this bit actually works.)
Not to mention you would also need energy to power the nuclear reactors in each design.
So really, we use chemical based propulsion systems because the alternatives are technologically expensive and difficult. We will struggle to make the chemical propulsion systems propel more because of the multiplying fuel issue (unless we find more efficient fuels). But really the biggest issue isn't so much the propulsion, its the distance!
For example the space station currently orbits us at about 18,000 miles per hour, orbiting the earth once every 90 minutes.
The Apollo spacecraft which flew to the moon traveled faster than that, at about 24,000 miles an hour. These types of speed are inconceivable for travel on earth, being hundreds of times faster than any jet can go.
So really my argument rests on these points, fuel problem, lack of easy alternatives, cost of fuel, sheer distance.