Late Roman and early medieval shields were round, and that style continued to be popular through the Viking Age in western Europe. However, around the 10th century, these shields started to grow longer - probably to cover the legs of horsemen. Here, for example, is a 10th century chessman on horseback. You can see how his shield extends the length of his leg, giving him full-body coverage on his left side. These shields were suspended with a shoulder strap as well as straps for the forearm, leaving the left hand free to hold the reigns.

This shape - the 'kite' shield - remained popular through the 14th century. By then, however, armor was becoming more sophisticated, and gradually horsemen relied more and more on their armor with shields growing smaller and becoming more maneuverable. The style of shield you link resulted from this shrinking process.

There are a few practical designs to the style that emerge from using one in modern re-enactment fighting. The corners on top help to protect your head, and the extended tip offers good coverage for your sword arm. Corners on the bottom of a rectangular shield are less useful, and add unnecessary weight; the triangular point is enough on the longer kite shield to protect the whole body when you fight on foot (your legs are narrow, and provide little target if you stand with one foot pointed toward your opponent and the other foot further back). The shorter triangular shield in your picture offers less leg protection, but legs are very difficult to hit in foot combat as an opponent must step far into your striking range to reach your legs, a dangerous risk. And on horseback, the legs were well armored.

You're correct that, for forming a tight shield formation like the Roman testudo there's nothing as good as a rectangular scutum, but that style of fighting was very specialized and even the Romans did not rely on it for most of their empire, preferring more flexible formations. In the later middle ages when these triangular shields were most common, it would be used by horsemen, while massed infantry formations used two-handed pikes/spears instead of the early imperial Roman shield, short sword, and javelin combination. For armored horsemen, the smaller shield was sufficient.