Our Gemara on Amud Aleph quotes a teaching from Rabbi Yishmael:
Sin obstructs and clogs the heart of the sinner.
On a simple level, the Gemara is noting that the momentum of sinful behaviors makes a person more callous and less sensitive to wrongdoing and morals in general. Presumably, this is the Timtum halev that the rabbis meant. A clogging of the heart and a contraction of open-heartedness and sensitivity.
Arvei Nachal (Bereshis 3) says a delicious peshat as to why this is true, and why Hashem would allow this to occur. He says that a person must derive life force to stay alive. Naturally, this life force comes from God. However, when one sins, he loses this life force and connection. If Hashem allowed this state of affairs to continue, the person would eventually whither and die as a result of loss of life power. Therefore, Hashem allows the Sitera Achera, the demonic evil forces to fill this space, to keep the person alive. However, the price to be paid is that as a result of having an ersatz and “non-factory grade” replacement part, there is an obstruction and barrier within.
We see this is true in so many ways, even in the physical world. For example, if you are used to eating junk food, regular food does not taste as sweet. If you go off sugar for a few weeks and you bite into a strawberry or eat an orange, you are overwhelmed at how sweet it is.
Addicts of any kind will tell you that once they leave their addiction, they notice fulfilling and subtle ways that their lives become filled, and they become high in a natural way. They can enjoy a hug from their children, holding their spouse’s hand, or gazing into their eyes, and feel a closeness and intimacy that eclipses, by far, the dopamine rush from their addictive substance or process. Even a person who is stuck in a bad behavior pattern, such as impatience or anger, can see a big difference in subtleties once this is overcome. Once the person goes beyond white knuckling and really learns how to sit with feelings, he or she will notice newness and beauty, even in the painful frustrations that normally would make him or her impatient and angry. Yet, it takes a willingness to be open and experience the subtlety. When one is steeped in fighting or avoiding problematic feelings, especially through addictive distractions, there is an inability to experience the sweet realness of life. This is what the rabbis meant by Timtum Halev.
Translations Courtesy of Sefaria, except when, sometimes, I disagree with the translation 
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