Why did my toes become dark purple?

It's really. Impossible to give you a diagnosis over the internet. A dark purple discoloration of the toes can come from a circulation issue, either a blockage or vasospasm (arteries that constrict causing less blood to flow), and sometimes, rarely, a small clot that breaks off from higher up in the body that occludes a small artery. It's important you see your doctor about this as soon as possible.
Many possibilities. There are many things that may cause a discoloration of the skin. Several types of circulatory and inflammatory conditions, allergic reactions, and sometimes bacterial, viral or fungal infections and cancer can cause a discoloration of the skin. Other thinks such as pigments from shoes or socks could be staining your toes. Have it evaluated by a professional and get the appropriate treatment.