First, protons and electrons are both particles. Waves at the atomic level are often referred to as probability waves (three-dimensional graphs that show the probability of finding an element at a particular point around a nucleus).
This is it. Let me start by saying that it is not necessary to see something to determine its properties. In fact, many properties can’t be understood by simply processing the light that bounces off an object. Although our brains are well-equipped to understand things in the world, based on their light emission and reflection, as well as transmittance, this is due to evolution. Low-light environments have allowed creatures to discover other methods of determining physical properties such as sound probing. As intellectual beings, we also have other methods to study our environment such as the electron microscope. Our conclusions are valid regardless of whether we use different techniques or apparatus to identify the characteristics of atomic particles.
So, why can’t we see protons and electrons? We have actually seen atomic nuclei, which are composed of protons, using electron microscopes. Because electrons are smaller than many atomic nuclei we can use them to see the features of atoms. However, it is impossible to see an electron. The electron is extremely small and has a very low mass. They are extremely fast and their exact position is unknown due to the Heisenberg Uncertainty Principle. Their exact position can be approximated to a certain degree of Uncertainty. It is impossible to see them because we cannot obtain an exact position. Their mass is so small that any interaction with them (involving an electron or photon), will send them flying away. We cannot see them, and have a worse understanding of where they are. The HUP makes it impossible to see electrons. This is why we now use the electron cloud model of atom. We only know where electrons will be and not where they are actually located. The electron cloud is a probabilistic field.
Again, sight is only one form of property detection. Therefore, the fact that you can’t see these things shouldn’t be a reason to be skeptical. We know electrons from more reliable sources than the photon detectors attached at image processing computers (i.e. your eyes and brain).