Why does an increase in thermal energy lead to an increase in entropy?

Question

After reading The Preference for Low Energy States, where it is stated in the excellent accepted answer that

As macro states with a lot of energy stored in heat (our ball with random thermal motion of its molecules) contain many more micro states and are therefore much more likely, energy tends to get transferred from potential energy to thermal energy. This is observed as a tendency towards a decrease in potential energy

I started to wonder exactly why this is the case. From What exactly is entropy?, I think that I got a basic good understanding of what entropy is. It seems to me that the statement from the quote is directly linked with Entropy, but I cannot figure out why.
I am primaily asking for an intuitive, non-mathematical (even though I obviously cannot say for sure if this will be possible to answer this question) understanding.
Why does an increase in thermal energy lead to an increase in entropy?

Possible duplicate: Does entropy increase with a decrease or with an increase in a system’s temperature? I don't think this answers my question because there is no explanation for why an increase in thermal energy equals an increase in entropy.

gandalf61 · Accepted Answer

A decrease in potential energy does not necessarily increase the temperature throughout a system, but it does move the system towards equilibrium.
If a container of gas is divided into two halves at different temperatures, this represents potential energy. If we allow heat to pass from the hotter half to the cooler half, and maybe use some of this heat to do work, then we reduce the potential energy of the system. Eventually the two halves reach the same temperature and are in equilibrium.
There are more micro-states in the equilibrium macro-state than in the original non-equilibrium macro-state (in the same way as there are more ways to toss a hundred coins and get fifty heads and fifty tails than there are to get forty heads and sixty tails). So the decrease in potential energy leads to an equilibrium state, which means an increase in entropy.

Why does an increase in thermal energy lead to an increase in entropy?

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