One of the biggest things that DD helps to do is to transform self-pity, which is a small mind, and helps us to get in touch with the bigger, expansive mind of compassion. When we are lost in self-pity we are thinking about ourselves, we have a lot of me, me, me. And we have a lot of limitations that come from only thinking of ourselves. So, "how can it support the growth and deepening of my practice?" If we let go of a certain amount of self, attachment to me, and open to a bigger mind, it can help to strengthen right intention, which is the second factor of the Noble Eightfold Path. Right Intention, which are thoughts directed towards harmlessness, free from ill-will and free from cruelty. So getting in touch with the bigger mind of compassion, which understands the universality of Dukkha helps us to overcome our ego, selfishness, and helps to understand a bigger view.
It helps us to get more Right View, understanding that we are not the only one who suffers. Dukkha is part of existence. So it helps to strengthen wisdom, Right View and Right Intention. If you strengthen wisdom, Right View and Right Intention, you have stepped onto the Noble Eightfold Path, and that is going to help you grow and deepen your practice. Over and over in the scriptures the Buddha used the universalizing method. For example, the Five Daily Recollections - "I am of the nature to decay, I am of the nature to be diseased, I am of the nature to die, all that is mine dear and delightful will change and vanish, I am the owner of my Kamma, etc." After he talks about this, he asks us to reflect that not only us but also others are of the nature to decay, be diseased and die, and that they are owners of their Kamma, heirs to their Kamma, etc. Then he asks us to reflect, not only in the past but also in present and also in the future all beings are of the nature to decay, be diseased and die, and that they owners of their Kamma, heirs to their Kamma, etc. So he taught us to see what we are subject to and see that it is universal. This helps us to start to understand the laws of existence.