Within the context of the Upanishads and Vedic thought in general, we are prompted to consider that there are not two birds in the tree but two aspects of the selfsame bird.
This bird is the 'individual self' that lives and acts and moves in the tree (the world), indulging in its fruits, taking the pleasures and pains that come as an inevitable consequence of action, agency and desires.
But this bird is also the bird that watches, is an unmoving witness to all that is passing, ephemeral, illusory. It is free from action, free from desire, free from ego, selfless.
This Upanishad is inviting us to feel, to experience, a distinction between the 'individual self' (the busy, 'doing' bird) and the 'universal self' which it calls 'pure consciousness'.
When we consider our meditation practice, our 'mindfulness' practice, are we not connecting in some way with this ancient wisdom that, by most reckoning, predates Buddha, Confucius, the Tao Te Ching, Socrates, Plato and Jesus, let alone the psychology and neuroscience of recent times? Here in this ancient text we are reminded that we are more than 'human doings', we are 'human beings', we are more than a perpetual waterfall of activities and concomitant emotions. We are more than our thoughts. We can sit still and watch our thoughts! We can watch the perpetual waterfall.
We may flit from branch to branch, seeking, avoiding, attempting to satisfy a multitude of appetites, but there is also the branch where we are still, where we witness the ebb and flow of a life that has the quality of Prospero's 'insubstantial pageant, melted into thin air'. We are indeed 'such stuff as dreams are made on', yet there seems to be something that doesn't 'melt into thin air', something that stays safely with us as awareness finds harmony with it. Here is found peace, joy, resilience, and freedom.