We see beautiful Catholic buildings adorned with multiple works of stained glass art; is this for any particular reason other than the fact that stained glass art looks nice?
Does Stained Glass art hold any special meaning for Catholics?
The short answer is yes.
For centuries, stained glass art in church windows, was known as the Poor Man’s Bible
The term Poor Man's Bible has come into use in modern times to describe works of art within churches and cathedrals which either individually or collectively have been created to illustrate the teachings of the Bible for a largely illiterate population. These artworks may take the form of carvings, paintings, mosaics or stained-glass windows. In some churches a single artwork, such as a stained-glass window has the role of Poor Man's Bible while in others, the entire church is decorated with a complex biblical narrative that unites in a single scheme.
The Biblia pauperum
The term Poor Man's Bible is not to be confused with the so-called Biblia pauperum, which are biblical picture books, either in illuminated manuscript or printed "block-book" form. The illuminated Biblia Pauperum, despite the name given in the 1930s by German scholars, were much too expensive to have been owned by the poor, although the printed versions were much cheaper and many were probably shown to the poor for instruction.
But despite the fact that the books, at least in their earlier manuscript versions, were created for the rich, while the carvings and windows of a great church provided free entertainment and instruction to all who entered the doors, there were strong points of similarity in both subject matter and iconography.
Reproduction of motifs
In a world before the printed book, fidelity to the original in transcribing of books by hand was the only thing that maintained the Bible and other works of literature for posterity. Along with the written words of the document were often transcribed commentaries and illustrations. While talented illuminators added their own style and embellishments, the form of many pictures remained the same, and different scenes or motifs were repeated many times and in different media.
There is, for example a particular motif of several sheep, one of which has a foot raised to scratch its ear, which occurs in Italy in the 13th and 14th centuries in manuscript illumination, wall paintings and carved stone panels. A motif of paired flying winged figures which is seen on pagan Roman sarcophagi passed into Christian art as a very commonly used portrayal of angels. The reproduction of figures from manuscripts was particularly common in stained glass windows with various Biblia Pauperum being frequent sources.
I had learned in Renaissance art history that stained glass was used in Flemish, English, Burgundian and Germanic churches to provide warmth (or at least not oppressive heat). Whereas in southern churches, you'd see a lot more panels that could be lifted to allow for ventilation. In modern churches this doesn't matter nearly as much. We open up stained glass windows.
Khan Academy has a good write-up on stained glass history. It mentions some of the things to look at in the stained glass, the techniques that light was used to evoke certain reactions and the evolution of the process of creating it.
The other part is (and this is true in American churches) you'll see patrons mentioned in the stained glass. In old frescoes and stained glass art you'd see patrons in the stained glass. But it's quite a bit more manufactured nowadays and not composed of thousands of intricately laid glass pieces, so the patronage would probably just be dyed in to a little scrollwork near the bottom.