Actually the answer is pretty simple: Symmetry furnishes organisms an advantage, but this can not be generalized. The sources you mentioned don’t have sufficient knowledge about biology and are not in the picture so they make their claims on the basis of humans and some limited number of animals they see around them.
For instance, sponges (Porifera) living on the bottom of the sea are completely asymmetrycal. Another example is Uca pugnax, a crab which has one chela more than 10 times bigger then the other. Yet another example is the Narwhal, which can have one extreme long horn whereas the other is tiny. One of the most interesting examples would be a species of the flatfish. This species lives on the bottom of the sea and the eyes of this species evolved asymmetrically. Basically they are like Rajiformes (ex: stingrays) which has the eyes on the side of their body. Despite their similar movement, the Flatfish has one big eye on the top of its head while the other one is small and on the bottom of its head. The reason for this situation is that the predators attack from the top and the food is on the bottom and the Flatfish does not need a big eye to detect the food. Similarly, the ears of most owls are asymmetrical. Despite our symmetrical appearance, our organs are definitely asymmetrical: the heart is on the left, the lungs are asymmetrical, the liver, the spleen and the intestines are all asymmetrical.
Symmetry, basically grants an advantage to the animal, the odds that danger or a pray is on the right or left, above or under are more or less equal. This is why organisms are mostly symmetrical. Again symmetry can be useful in terms of physic, an asymmetric cheetah might not run fast enough or an eagle might not fly efficiently (vide: Friction, Moment).
Sometimes nature can annuliate symmetry with enviromental factors and according to statistical data, being completely symmetrical is impossible. The best known example is our face. The length of your arms can vary even if for millimeters. Currently two members of the Tree of Evolution (a branch of METU Biology and Genetics Student Club) are working with Assoc. Prof. Meral KENCE in the Biological Sciences Department of METU on the subject of “Fluctuating Asymmetry”. It is possible to find the evolutionary history and relevants of species based on their drift of symmetry or, in other words, their asymmetry. As it can be seen from this example, evolution can be supported from billions of different aspects.
Whether an organism is symmetrical or not is furnished by the genes, as usual. Your arms and legs are symmetrical on both sides of your body because it is coded like this in your genes, but this can change due to growth disorders. For example, some people develop asymmetrical legs leading to disformations of bones which again can lead to waist and back aches. Genetic disorders and mutations can cause such asymmetries as well.