An important concept in quantum physics are standing waves. They help describe a large amount of phenomena that physicists struggled to explain classically. This includes the aforementioned black body radiation or the orbit of electrons. While it is too complicated to explain here how exactly they are used in quantum mechanics, the general concept and why they are important can be explained. First, the general concept of standing waves is relatively easy to grasp. When you have a string and send a wave down it, it will eventually be reflected back. If this is done at the correct frequency, the waves and their reflection interfere to result in standing waves. They consist of nodes and anti-nodes.
(source: https://www.slideserve.com/deanne/wave-properties-e-m-spectrum)
All nodes remain still throughout the process, while the anti-nodes oscillate up and down rapidly. The amount of standing waves is directly proportional to the frequency.
The standing waves you came across, while solving the room were, however, not only on a one dimensional line, but on a two dimensional plate. This plate is called a Chladni Plate. In principle though, this works the exact same as in the one dimensional case. A rod in the centre of the plate is oscillates up and down at a certain frequency. This generates waves on the plate. At the edge, they are still reflected back, just as in the one dimensional case, which can result in standing waves. The shape of them depends on the shape of the plate’s border. These standing waves are also only generated at certain frequencies.
The reason this is such an important concept in quantum mechanics comes from the math that underlies quantum mechanics. The entire field heavily relies on the mathematics of linear algebra. In this field of math, operators, represented by matrices are of vital importance, however, these often have something called eigenvalues and eigenvectors. Standing waves can be seen as the eigenstates of these oscillating systems. Thus, in a theory relying on this math it is intuitive that these types of waves plan an important role.