The importance of the cutting down of the boughs, as seen in Act V, Scene IV of Shakespeare's Macbeth, refers back to the prophecies of the apparitions in Act IV.
The Third Apparition tells Macbeth the following:
Be lion-mettled, proud; and take no care
Who chafes, who frets, or where conspirers are:
Macbeth shall never vanquish’d be until
Great Birnam wood to high Dunsinane hill
Shall come against him.
Therefore, when Malcolm orders the soldiers to cut down the boughs, he is unknowingly referring to the fact that Macbeth's demise will come from Birnam Woods.
Let every soldier hew him down a bough
And bear’t before him: thereby shall we shadow
The numbers of our host and make discovery
Err in report of us.
Earlier in the text, Macbeth refused to believe in the prophecy provided by the Third Apparition. He stated that the woods could not fight against him, stating that the trees could not pull their roots from the ground and fight.
That will never be
Who can impress the forest, bid the tree
Unfix his earth-bound root?
The fact that Malcolm tells the soldiers to cut down branches and use them as cover speaks to the truth of the prophecy.