How do we know fusion energies? Crucial to our understanding of stellar evolution is the energies involved in nuclear fusion reactions.
How do we know those energies? How do we know which fusion reactions take place, at which energies, and which is exothermic or endothermic? Have we been able to actually conduct these reactions in accelerators (or some other apparatus)? Or does our knowledge of these energies come from theoretical considerations? Or both? 
And what theory tells us what the fusion energies will be?
 A: The overall energy yield of the reactions is not so difficult to measure - it simply amounts to measuring the masses of the nuclei and using the Einstein $E = mc^2$ formula - subtract the mass of the nucleus from the sum of masses of protons and and neutrons within it and that is your binding energy. Experimentally this is done using the methods of mass spectrometry. In the end, you get a fully empirical plot for binding energies that looks like this

(Source: wiki)
The measurement of fusion cross-section is much more involved, though. On a theoretical level we know that the two ions need to 1) get over the Coulombic barrier (imagine two positively charged balls trying to hit each other), 2) not to "tunnel through one another" (this has to do with quantum mechanics and their DeBroglie wavelength), and 3) overcome certain finer barriers on a nuclear-nuclear level. From that, first models that will give you correct ballpark numbers for cross-sections can be formulated. However, in a real plasma you also have to account for other effects such as screening by electrons. 
Information can also be gained from well understood astrophysical objects such as the Sun and solar neutrinos, whose rates and energies, along with high-fidelity models of the solar interior, constrain fusion cross-sections of light elements. Another interesting example of nuclear physics getting information from astrophysics is Fred Hoyle's 1953 prediction that there must be an additional resonant channel for the fusion of Carbon, because otherwise it would not be so abundant.
Other information can be gleaned from particle-collider experiments. If you take a beam of ions, sufficiently accelerate it with an electrostatic potential, and hit a target or another beam, it is not too hard to reach fusion energies or at least energies that constrain the interaction model even in a medium-sized lab. (What is difficult is translating the results of these experiments into reliable fusion rates/cross-sections.) 
A: 
How do we know those energies?

Thanks to Rutherford's lab. In 1934 a group led by Mark Oliphant shot protons at lithium foils, to which they later added deuterium from heavy water. Every so many million of such shots would result in a high-energy alpha, which they could see on a screen. By varying the driving voltage and measuring the output energy they could calculate both the threshold energy (the Coulomb barrier) and the resulting energy release of the reaction. This work revealed the isotopes of hydrogen.
Variations on this basic theme have been used continually since then, to test different fusion cycles and refine the measurements. This was a particularly active field in the 1970s.
