Birnam Wood comes to Dunsinane because Malcolm's soldiers carry branches cut from it in front of their bodies. The witches' promise is fulfilled by a field order.
The promise comes from the third apparition (Act IV, Scene 1), a crowned child with a tree in his hand: "Macbeth will never be defeated until Great Birnam Wood marches against him, covering the twelve miles to the heights of Dunsinane Hill." Macbeth finds it impossible: "Who can recruit the forest and order the trees to pull their roots loose from the ground?" In Act V, Scene 3, he leans on it: "I will not fear death or ruin, for Birnam Wood has not yet come to Dunsinane."
The army of Malcolm, Siward, and Macduff reaches Birnam in Act V, Scene 4. Malcolm orders: "Let every soldier cut down a branch and carry it in front of him." The reason is to hide the size of the force, so that the enemy's informers miscount.
In Scene 5, a Messenger swears he saw the wood begin to move from the hill. Macbeth answers: "I begin to suspect how the devil plays with words, lying as if he spoke truths." In Scene 6, in front of the castle, Malcolm orders the branches thrown down, and the army shows itself. In Scene 7, even knowing the wood has arrived, Macbeth decides to fight to the end.