There are several different ways to look at it. One definition of "entropy" is the thermodynamic definition:
##\Delta S = \dfrac{\Delta Q_{rev}}{T}##
where ##\Delta S## is the
change in entropy and ##\Delta Q_{rev}## is the heat added to the system (or removed, if ##\Delta Q## is negative) and where ##T## is the temperature (in Kelvin). (The subscript "rev" means "reversible"; basically, it's the heat that would be added if you had made the change slowly and reversibly.)
So if you transfer heat from a hot object (one with a large value of ##T##) to a cold object (one with a small value of T), then the entropy change of the hot object is negative, because it's losing heat, so ##\Delta Q## is negative. The entropy change of the cold object is positive (because ##\Delta Q## is positive, since it's gaining heat). But since entropy is inversely proportional to temperature, the amount of entropy lost to the hot system is smaller than the amount of energy gained by the cold system.
So saying that entropy always increases boils down to saying that if you put a hot object into contact with a cold object, the hot object gets cooler and the cold object gets warmer. If you want to know why THAT always happens, you've got to go a little deeper.