The COVID causing coronavirus invades the cells by attaching to a special receptor called Angiotensin converting enzyme (ACE-2). The ACE 2 serves as a gateway for the entry of the virus into the cells following which the virus begins to replicate itself and invades more cells.
In the human body, the ACE2 receptors are found in many organs including heart, stomach and even skin. So far, this is considered to be the best explanation on why COVID affects other body organs despite being a respiratory tract infection.