Solid state is characterized by solid components being very close one to each other (strong attraction forces), liquid state is characterized by medium distances between liquid components (medium attraction forces) and gas stated is characterized by large gaps between gas components.
To change the state of a body, from solid to liquid or further from liquid to gas, means to give its components enough vibrational and translational energy that the bonds between them will break. The components that make the solid, liquid and gaseous states above are molecules, and the interaction forces that defines these states are the intermolecular forces. Molecules themselves do not break separately into atoms except in chemical reactions. Thus, not the strength of covalent bond (which is a chemical bond) is important but the intermolecular forces. Since as you say, covalent compounds have weak intermolecular forces, their melting and boiling points will be low (compared for example with ionic compounds). Smaller energy will be necessary to be transmitted to oppose the forces acting between molecules.