In a number of ways. Let’s start with the second noble truth, that craving is the cause of suffering. If suffering is to end, craving needs to be eliminated. Yet although the Buddha gives instructions on how craving can be reduced, it is only when you eliminate delusion, avijjā, that craving is altogether eliminated. This deeper aspect is seen when DO stands in for the usual phrasing of the second noble truth, for instance in AN 3.61. This adds an important dimension to this truth, for without it it would not be clear how you could make an end of suffering. The elimination of delusion, of course, is a core aspect of what meditation is all about, as in samādhi being the condition for seeing things according to reality.
The above might seem to belong to theory, not to practice, but I don’t think it is always possible to distinguish between the two in absolute terms. You need a proper framework for the practice to make sense. And you need a framework that expresses the Dhamma in such a way that people practice it all the way to the end of the path. If you don’t have a complete picture of the path, the real Dhamma will soon be lost.
There are number of ways in which DO provides such a theoretical-cum-practical framework. I will only be able to give you a taste of the possibilities, since this is really an enormous topic.
For starters, the sequence avijjā-saṅkhāra-viññāṇa (delusion-intentional activities-consciousness) provides an in-depth analysis of the causes and results of kamma. In brief and much simplified, because of avijjā, we mistakenly think we should pursue worldly happiness and we see ourselves as agents capable of procuring such happiness. Because of this cognitive distortion, we set out to secure this happiness for ourselves, which is what saṅkhāra, intentional activities, are all about. As we do so, and depending on our commitment to morality (which is also affected by our degree of avijjā), we engage in activities that are both wholesome and unwholesome. The sum of these actions affects our mind/consciousness (viññāṇa). If we predominantly do good, we feel good/better about ourselves and our mind brightens up. If we do a lot of bad, it will have the reverse effect. In this way we are stationing our consciousness at a particular “level”, a level that will tend to continue when you pass away. In this way, the limits of nāma-rūpa are set for your next existence, which is equivalent to being reborn in a particular realm. This, then, shows how intentional activities are the result of delusion, and how these activities in turn affect our minds and therefore also our rebirth.
Then there is the interesting mutual conditionality between viññāṇa and nāma-rūpa, consciousness and name-and-form, as seen in DN 15. This is in many ways the critical insight the Buddha had into the non-self nature of the mind, which provided the basis for his rejection of the Vedic eternalist philosophy. The way this mutual conditionality provides a middle way between eternalism and annihilationism is beautifully set out in the Kaccāna-gotta Sutta, SN 12.15.
Then there is the linkage between consciousness, name-and-form, the six sense bases (salāyatana), and contact (phassa). DN 15 has interesting variation on these four factors, where the six sense bases are left out. The Buddha then proceeds to show how contact is dependent both on the physical impact of an object on the sense organ (paṭigha-samphassa) and on the mind’s interpretation of that sensory input (adhivacana-samphassa). This shows nicely how our perceptions of the world are in large part fabricated by the mind. As we develop our minds, the world changes.
Then there is the process of craving, uptake, and existence (taṇhā, upādāna, and bhava). Craving makes you take up various pursuits: getting an education, getting a job, finding a partner, living in a house, becoming a Buddhist, etc. You take these things up because they help satisfy your cravings. But the effect of taking things up is that your life takes a certain shape, you exist in a certain way. You can tell the nature of your own existence by listening to your mind and seeing what it is preoccupied with. For most people, their preoccupation will be related to the sensual realm. As your meditation deepens, your interests will change accordingly.
These are some of the interesting psychological insights that can be gleaned from DO. All of them have real, experiential equivalents, all of which are useful, often very useful, on the path.
Many of the above linkages can be experienced in a fairly straightforward way. Take the first three factors: you can easily see how your pursuit of happiness in the world gives rise to wholesome and unwholesome actions, which then have direct effect on how you feel about yourself. What you are seeing is the kammic link between intentional activities and feeling. Once you see this clearly, you get a powerful motivation to live well.
If what you mean, however, is the full insight into the core mechanism of DO, that is, how craving gives rise to suffering/rebirth, then this only happens with streamentry. Streamentry is defined as the full insight into the Dhamma, including seeing the five khandhas (the five aspects of personality) as suffering. When you see this, all tendencies to crave will cease (temporarily), because you cannot crave for suffering. When you see the full ending of craving – like a frog seeing water when emerging from it for the first time – you also understand its power to keep the round of rebirth and re-death going indefinitely.
Samādhi is enough. Insight into previous lives is not the same as seeing DO. In other words, you can see your past lives and still lack this insight. Insight into DO is insight into causality, that is, as long as you crave you will continue to exist, including being reborn.
Have I answered your questions? Have I made any sense? Good communication is just so hard!