I have heard that on one occasion, the Blessed One was staying at Uruvelā on the bank of the Nerañjarā River at the root of the Bodhi tree — the tree of awakening — newly awakened. And on that occasion he sat at the root of the Bodhi tree for seven days in one session, sensitive to the bliss of release. Then, with the passing of seven days, after emerging from that concentration, in the first watch of the night, he gave close attention to dependent co-arising in forward order, thus:
When this is, that is.
From the arising of this comes the arising of that.
In other words:
From ignorance as a requisite condition come fabrications.
From fabrications as a requisite condition comes consciousness.
From consciousness as a requisite condition comes name-&-form.
From name-&-form as a requisite condition come the six sense media.
From the six sense media as a requisite condition comes contact.
From contact as a requisite condition comes feeling.
From feeling as a requisite condition comes craving.
From craving as a requisite condition comes clinging/sustenance.From clinging/sustenance as a requisite condition comes becoming.
From becoming as a requisite condition comes birth.
From birth as a requisite condition, then aging-&-death, sorrow, lamentation, pain, distress, and despair come into play. Such is the origination of this entire mass of suffering & stress.
Then, on realizing the significance of that, the Blessed One on that occasion exclaimed:
Notice that dependent co-arising (paṭicca samuppāda) is expressed in terms of processes — of events and actions — without reference to a framework containing those processes. In other words, it doesn't mention the existence or non-existence of agents doing the actions, or of a framework in time and space in which these processes happen. Thus it makes possible a way of understanding the causes of suffering and stress without reference to the existence or non-existence of an "I" or an "other" responsible for those events. Instead, the events are viewed simply as events in the context of the process — a way of viewing that makes it possible to abandon clinging for any of these events, so as to bring suffering to an end. Even the idea of an "I" or an "other" is seen simply as part of the process (under the factors of fabrication and the sub-factor of attention under "name" in name-and-form). This is what makes possible the abandoning of any attachment to the conceit "I am," as mentioned in Ud 2:1, 4:1, 6:6, and 7:1. In this way, the treatment of dependent co-arising in the first three udānas, while terse, actually sets the stage for understanding some of the more paradoxical teachings that appear later in the collection.
For a discussion of dependent co-arising in general, see The Shape of Suffering. For further discussion of its role in framing and abandoning thoughts of "I am," see Skill in Questions, chapters 3 and 8.