They do not tell us about the interior directly. It is possible to determine the temperature and pressure of the surface from the spectrum. After that, one has to resort to modeling of heat flow, pressure balance, nuclear reaction rate and knowledge of the sun's mass (from planet speeds)to come up with conditions at the center.
The pioneer of this process was S. Chandrasekhar, who won the 1983 Nobel Prize for his work that he had done about 40 years earlier. The Nobel Prize committee is often late to recognize great work.
There is a newer technique (developed in the last 20 years) call helioseismology that can also be used. It measures oscillations and modal patterns in the local in-an-out "drum-head like" vibrations of the solar surface to diagnose the variation of density with location inside. Precise Doppler shifts of atomic lines are used for the velocities. The math is quite complex.
For more about it, see
http://soi.stanford.edu/results/heliowhat.html
How do observations of he sun's surface tell us about conditions in the solar interior?
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