What is the meaning of 'awareness of the flow'?
When I read your question I actually think you gave a perfect answer, i.e. "...the continuous flow of arising and passing away of all conditioned things...".
I'm not too familiar with Stephen Levine or his book but it sounds like he is teaching about exactly this quality of conditioned phenomena. What is this quality?
Conditioned phenomena is also called compounded phenomena, meaning that they consists of other phenomena that again consists of other phenomena. In other words, we are here dealing with phenomena based on causality.
In Buddhism we have a Dhamma called the Three Characteristics of Existence, which are impermanence, unsatisfactoriness and uncontrollability. These characterize conditioned phenomena. In regard to the question, especially impermanence and uncontrollability is important here.
Impermanence shows that phenomena arise and cease. They come and go according to their respective causes and conditions. Uncontrollability shows that there is no stability or control to be found anywhere. One cannot exercise control over this flow. Mahasi Sayadaw calls it therefore oppressive and terrible. Oppressive is the incessant arising and ceasing of conditioned phenomena.
It sounds very much as what Stephen Levine is teaching about and what you yourself are describing. Now, the Buddha once said that its okay to use conventional language but that one should not be fooled thereby. In this regard, it means that all these different words (concepts) really just mean the same thing. Its just different ways of teaching it.
If we really want to understand what flow is then the practice of insight meditation should be undertaken. Intellectual knowledge only goes so far. With wisdom gained from insight meditation one will be able to see conditioned reality with ones own eyes.
In the Sabbasava Sutta (MN 2) the Buddha taught about 7 taints that should be overcome and with what method they should be overcome. The first taint should be overcome by seeing, i.e. The Four Noble Truths should be seen with ones own eyes. The truth about Samsara should be seen with ones own eyes, in here lies the incessant arising and ceasing of phenomena.
By gently letting go of everything—not through force, not by slaying it, but simply seeing all the content as passing show, as process and flow—we become the whole of our experience and open to our natural understanding.
It might mean that when staying in the present moment and not letting the mind straying into past or future even for one moment, Wisdom (natural understanding) will arise.
We can observe [with our practice] what anger feels like, what joy feels like, what separation from the flow feels like, what fear or worry feels like.
Separation from the flow, meaning that when the mind has wandered off into past or future one disconnects from ultimate reality and instead ventures into conceptual reality.
What joy, fear or anger feels like, means to know the quality of that object when it arises, as it exists and when it disintegrates. To thoroughly contemplate the objects qualities and distinctive marks.
[Hindrances] distract us from an even-minded awareness of the flow.
When hindrances are present one cannot clearly see reality as it has been plastered over with many different colors of dye (example given by Ven. Bhikkhu Bodhi).