Explain why molecules held together by covalent bonds don't have electrical charges?

Because in a covalent bond, the electrons are shared within the molecule. There is no lone electron that makes the molecule to be charges. Remember that to conduct electricity, ions should be used because they have charges. Charges are brought about by incomplete or have additional electrons. But with a covalent bond, the electrons are already happy in the molecule that no one is left out to look for another one to attract to.