This is what it says on the linked page:
...thermal reversibility requires that all heat transfer is isothermal.
This statement is only true in the context of that web-page, where it is being assumed that the system is in contact with a thermal reservoir (i.e. an object whose thermal mass is so large that it's temperature doesn't change while exchanging energy via heat with the system).
We know that a process in which there is energy transfer via heat between two objects at different temperatures is irreversible, and so in order for a process to be reversible in this context, the system and the environment must be at the same temperature at all times during the heat transfer. Since the environment's temperature doesn't change, neither can the system's.
A reversible process in which energy is exchanged via heat between a finite system and an infinite reservoir must be isothermal.
Now, it is possible to design a reversible process in which the system's temperature changes and the system exchanges energy via heat with its environment. If you bring a system into contact with a sequence of thermal reservoirs, each at an infinitesimally larger temperature than the last, and you allow the system and reservoir to equilibrate at each step, then the system gains energy via heat, and its temperature goes up. Crucially, this heat exchange occurs when the system is in contact with a reservoir at the same temperature, so the entire process is reversible.