How to accurately measure the temperature of red supergiants?
Stars behave like blackbodys. Not perfect idealized blackbodies, however, the spectrum of a star is close enough to the standard blackbody spectrum. Reason why you can use the Wien's Law to calculate an estimate of its surface temperature:
$\lambda_{\rm max} = (0.29 {\rm\, cm\, K}) / T$
Where $\lambda_{\rm max}$ is the frequency of maximum measured emission of light. Problem is, measuring a star’s spectrum is not always easy. The most common approach is to observe the star with two different filters, a blue one and a visual yellow one. Measuring the difference of the Blue bigthness and the Yellow one gives you the actual color of the star. In some cases more color filters are used (UV, Near Infrared, V, ...).
Now, as of why the used the iron lines: When a Star ages, its core gets more and more iron rich due to nuclear fussion (iron is the last element that produces more energy when getting fussioned than it takes for the reaction to happen). When the balance between outwards radiation and gravity gets unstable, the star collapses. In the case of Red giants a supernova most likley follows. The Red giant that was beign messured in the paper had probably a lot of iron, so the spectal lines of iron where the most &quot;stable&quot; ones that did not get altered by the stars upper atmosphere. They probably used two frequencys of different iron configurations to propperly estimate the $\lambda_{\rm max}$ of the star.
Hope this clarifys things a bit.