"Integration" is a very common term so it should be easy to read a better description that I'm about to give. My take on it is that integration is the process of reflecting on one's experience and figuring out what it means for your real life.
For instance, say someone is burdened with thoughts of self-hatred due to growing up with critical and non-supportive parents. Although they can intellectually entertain the thought that the self-hatred is due to this, under the influence of psychedelics, they might viscerally feel the truth of it. When sober again, integration means taking that insight and working on changing your mindset to align with that new belief.
There are plenty of examples of the flip side too, eg, someone with a string of bad relationships might, under the influence, really see that they themselves are the main cause of the failures. But an insight isn't enough. Integration is making changes in sober life to change things in the positive direction.
Or maybe your life is going great, you take psychedelics, and you have a hard trip that is filled with morbid thoughts about sickness and death. It can be interpreted as an indication that those thoughts are present but subconscious even in sober life, and maybe it is worth investigating more via conventional means.