3 Answers

Figure. Schematic of the 4 layers of the Earth based on chemical properties.

We know by both mathematical and empirical figures - physics, chemistry, and mathematics as John Morrow said. The inference is as such:

We need to know 3 things about the "center" of the Earth to deduce its temperature: the substance, physical state, and pressure.

1. The substance.
The substance is deduced when we recognize that 1) we have a magnetic field and 2) whatever is at the center of the earth must be denser than basalt (what our mantle is made out of), because, for one, if our planet was made solely out of basalt, the earth's calculated density would be lower than what modern day physicists have calculated. Iron just happens to be the most common substance that is both magnetic and more dense than basalt.

2. The physical state.
The physical state of the center of the earth further supports the 
calculations from the above figures. 2 types of waves are sent into the 
earth whenever an earthquake occurs: P-waves and S-waves. The 
characteristics of each wave are not so relevant to this question, but 
the important part is that P-waves travel through any substance, and we 
can observe seismic waves and tell how long it gets from point a-to-b, and how 
fast it moves through certain parts of the earth in its travel. Seismic waves happen to move faster in denser material, so we know that at a certain point deep in the middle of the earth the material changes from a less dense material (molten outer core) to a more dense material (solid inner core).

3. The pressures.
The ability for a substance to stay solid is related to temperature and pressure. One of the first things I learned in Geology class was that the higher the pressure, the higher the temperature required to melt that substance - i.e. the higher the pressure exerted on a rock, the higher the temperature that would needed to be reached to make that substance a liquid. If we can calculate the pressure (physics can, and that number is around 340 GPa or 3,400,000 atm) and know the physical state of the substance (solid iron), we can extrapolate from known physical states of iron at particular pressure and temperature levels to what we now roughly arrived at: depending on who you're talking to, and what exact figures are crunched, you arrive at 5000-7000 degrees Celsius. That is the temperature by which iron can stay solid under 340 GPa of pressure.

Certainly an error margin, but I know nothing outside of science that has given a better and more thorough answer.


citations:
everything else - Inner core
image - The Dynamic Earth @ National Museum of Natural History

Well, we can't "measure" it.  For obvious reasons, we can't get an instrument to the centre of the earth.

We can "calculate" it, using complex physics and math that I don't pretend to understand.  In the same way that we calculate the temperature of suns we can't visit or the gravity on planets we haven't been to.