Plant root cells, like all the other cells of the plant body, contain mitochondria and need to have oxygen to do the process of respiration to get energy. When a plant is growing in soil, the soil has to be loose enough to allow air to infiltrate so that gas exchange can occur across the epidermis of the roots. When you grow plants hydroponically, you need to bubble air into the liquid culture solution so that oxygen can dissolve from the bubbles into the water and diffuse into the root cells. This is the same reason most aquarium fish need to have an air bubbler hooked up to their tank.
Covering the container will help keep algae from growing in the culture solution. The liquid is designed to support plant growth, so any algae that get into it will flourish if there is light. By covering it you can prevent that from happening. The plant roots don't need light, because they are dependent on the photosynthetic upper shoots of the plants to do the photosynthesis and supply them with carbohydrates.