Hydroxychloroquine and chloroquine are zinc ionophores, a class of drug that carries the mineral zinc into the cell, without which free zinc would simply float around in the blood or be involved in other metabolic activities in the body. Quercetin and EGCG (green tea extract) are also zinc ionophores. These over-the-counter supplements exhibit similar pharmacological activity as the anti-malaria chloroquine drugs, and are much more widely available.
Zinc Ionophores are carriers of zinc thru cells' membranes, depositing zinc ions into the cytoplasm. NIH says that zinc ionophores specifically affect malignant cancer cells. With respect to coronavirus (and many other human RNA viruses using the same mechanism), zinc ionophores inhibit viral replication by affecting RdRP, RNA-dependent RNA polymerase, an enzyme used by the virus not unlike a lockpick, to hijack the protein factory within cells--the "translation process" whereby normal operations of the cell convert rna into proteins. This is how viruses take over the cells' protein factories to make RNA copies of itself.
The zinc inhibits this RdRp enzyme, inhibiting the virus from copying itself. During this time the virus is inhibited from replication, the immune system can destroy the unmoored and inhibited virus copies floating around outside of the cell, as well as the infected cells.