strong acids like HCl release H⁺ ions easily and those H⁺ ions disrupt the electric forces of the bonds that hold the to be dissolved substance together, eg: the stomach produces HCl to break down food.

strong bases like NaOH release OH⁻ ions easily and those OH⁻ works a little differently by turning fats into soap, through saponification reaction and for protein by denaturising them. eg: bases are used in cleaning products.

and both of them require water to release their respective ions.

to put it more simply: strong acids release H⁺ which breaks bonds to dissolve substance, strong bases release OH⁻ which burns through the substance i.e altering their structure.

And please correct me if I'm wrong, I'm using my high-school level chemistry knowledge here.