Let's first consider a white shirt. The individual fibers that make up a white shirt are really not that white. They are actually mostly transparent. However, transparent materials do reflect a small amount of light. For instance, a glass window is transparent (allowing us to see the outside world) but still reflects some of the light that hits it. This is why you can see your reflection in a window at night. Shirt fibers are the same. They are mostly transparent and slightly reflective at the same time. A single fiber therefore looks mostly clear, not white. However, each fiber is surrounded by more fibers which give the light more chances to be reflected. It's important to realize that the reflection of light happens at the interface between the fiber material and air. Therefore, a material that is made out of many small fibers has a large surface area, giving the light many opportunities to be reflected. Furthermore, the weave of the fibers creates many layers of reflecting surfaces. Therefore, the portion of the light that successfully makes it through the first fiber without being reflected can still be reflected by the lower layers. As a result, most of the white light falling on an undyed shirt eventually gets reflected back away from the shirt in a random fashion. The shirt is therefore seen as a diffuse white color. This also means that very little light makes it all the way through the shirt. We therefore see the material as opaque. The main factor that turns a mostly transparent material (the individual fibers) into a mostly opaque material (the shirt) is the array of multiple reflecting surfaces created by the weaving pattern.