How could we have an exchange of heat at constant temperature ? Isn't heat related to temperature ?
First of all i am going to concentrate on an ideal gas system.
Now when we say that during a process, the temperature remains constant, we are essentially talking about isothermal process.
Heat is just a form of energy, hence when we supply heat to a system and if all of that heat energy gets utilized in expansion of gas, that is in doing work for moving the piston by the gas, then no energy is used up in increasing the temperature of the system. Such process turns out to become isothermal.
Generally this is accomplished by providing heat very slowly and allowing the gas to expand such that temperature remains almost const.
Though heat is generally related to temperature, but we should think that heat is just energy and energy can alter any of the system variables of the system and not just temperature. If temperature is by design kept const, then the energy has no other way but to alter other system variables.
In addition to what @Nisargjoshi says, heat is energy transferred from one object to another due to a temperature difference in the objects. The energy flows from the warmer object to the cooler object. If the two objects have equal temperatures, there is no heat transfer. Energy would have to be transferred by some other means.