How is gene expression estimated?
The technique described here is called microarray. Your question has given me an opportunity to put forth one of my opinions about certain problems of gene expression studies.

Gene expression is a measure of the activity of any gene. If the gene performs its activity in the form of a protein, then its expression should be a measure of the protein. If a gene makes a non-coding RNA then its expression is a measure of the RNA concentration.

[You can omit the cases of post-translational modification because of signal transduction. They are highly expressed and transiently but frequently used. ] 

Like your example there are many studies which use mRNA concentration as a proxy for protein activity. This proxy works in many cases because transcriptional gene regulation is more frequently used mechanism for imparting stable changes. But the best strategy would always be to measure the proteins also. 

Apart from microarray there are several techniques to measure RNA concentration:


RNA sequencing (high throughput)
Real-Time PCR (medium throughput)
Northern Blotting (Low throughput, semi-quantitative)


Many techniques exist for measuring proteins also:


Mass spectrometry (high throughput)
ELISA (Low throughput, quantitative)
Western blotting (Low throughput, semi-quantitative)


Usually the "proxy method" is used because protein quantification is comparatively more difficult. Antibody based techniques like ELISA and Western blotting have problems of cross comparison of proteins because of the variability of the antibody binding efficiency.