How does the tree heal itself?

The tree in the poem heals itself by "absorbing years of sunlight, air, water." Even if attacked, hacked, and chopped, the tree will heal itself by drawing upon the earth and its surrounding environment. It will absorb the sun's energy and water from the soil.
The speaker says that the only way to kill the tree is to pull out the root, "Out of the anchoring earth." The implication, therefore, is that, if the root is not pulled out, then it will be through the root that the tree continues to heal itself. It will continue to absorb energy from the soil.
The whole poem is in part a celebration of the strength and endurance of trees. They survive despite being hacked at and chopped at, despite being scorched and choked. It takes a huge effort to kill a tree, which in turn implies an attack on those who make the effort nonetheless.

In this poem, Patel is emphasizing the fact that killing a tree is not something that happens accidentally—it requires a prolonged attack and is an effort of will. He illustrates this by describing the process by which "bleeding bark" will actually heal itself. This is a natural process by which trees seek to protect themselves from attacks. Trees have an active mechanism to try to prevent disease and injuries, and the purpose of bark is to protect a tree. At the same time, new shoots, "curled green twigs," will rise up from near the ground, so even if the original strong tree which people destroyed can never completely recover, the tree can "heal" itself by regenerating and creating new offshoots Eventually, then, the tree will "expand again" to the size it once was.
As Patel goes on to explain, the only way to fully kill a tree is to pull it out by its roots. He describes the action of killing a tree in almost gruesome detail, underlining the fact that this is a disruption of the natural process by which trees can survive most attacks. Trees are hardy creatures, and it is only through sustained and deliberate abuse that humans have succeeded in killing so many of them.