The reason we don't eat with our hands is to avoid pathogens. Specifically, the types of pathogens that spread via the fecal-oral route. These include protozoan parasites such as Giardia, bacterium such as E. coli, helminths such as pinworms (Enterobius vermicularis) hookworms, and tapeworms.
In countries where people eat with their hands, these parasites are endemic.
These parasitic infections are classified as "neglected tropical diseases" and the ignorance of the populations in tropical countries with regard to food hygiene is a major reason they suffer from these parasitic diseases.
We don't eat with our hands for the same reason we cook our foods - to avoid parasites.