How do you explain act 5, scene 7, line 55–60, of Macbeth?

There are only 31 lines in scene 7, so I wasn't sure if you meant scene 7 or scene 8. I've told you about both scenes here, because scene 8 is the conclusion of the actions set up in scene 7.
Act 5, the final act of Shakespeare’s Macbeth brings together many of the witches’ predictions from early acts. While some of the earlier scenes are longer, both the pacing and action of Act 5 are much quicker.
To understand scene 7, you must first remember the witches three statements in Act 4, scene 1. There, the three witches warned Macbeth to “beware the Thane of Fife” and promise him that “ne of woman born Shall harm” him. He hires murderers to kill the Thane of Fife’s family (he only is able to kill his wife and son) and is confident that he is safe, since naturally all people are born from a woman. The witches conclude their prophecy by telling him that his rule as king is safe because “Macbeth shall never vanquish'd be until Great Birnam wood to high Dunsinane hill.” Macbeth doesn't think that an entire forest can move, so he takes the witches at their word and continues his plan for power.
As Act 5 unfolds, things start to unravel for Macbeth. The Thane of Fife has convinced Malcolm to return from England with an army to fight for his right to the throne, and Lady Macbeth has committed suicide. In scene 7, Macbeth is trapped at the castle as the army closes in: "they have tied me to a stake; I cannot fly." However, he remains confident that he will be safe, quoting the prophecies to himself. The scene begins as he kills Young Siward who challenges Macbeth to a fight, but Macbeth again reminds the audience that he cannot be killed.

Thou wast born of woman / But swords I smile at, weapons laugh to scorn, / Brandish'd by man that's of a woman born.

As Macbeth exits, MacDuff (the Thane of Fife) arrives and demands Macbeth fight him. MacDuff wants revenge for the murder of his family.
In scene 8, everything makes sense at last. In this last scene, we learn that the witches' prophecy was correct: no one who was born naturally could hurt Macbeth. However, the witches have a loophole. The Thane of Fife was born early through a caesarean section and thus poses a threat to the king:

Despair thy charm; / And let the angel whom thou still hast served / Tell thee, Macduff was from his mother's womb / Untimely ripp'd.

In act 5, scene 8, MacDuff kills Macbeth and greets Malcom as the new king in lines 55–60.

Macduff. Hail, King! for so thou art. Behold where stands Th’ usurper’s cursèd head. The time is free. I see thee compassed with thy kingdom’s pearl, That speak my salutation in their minds, Whose voices I desire aloud with mine.

This is MacDuff officially formally recognizing Malcolm as the new King of Scotland and announcing that Macbeth, a usurper because he stole power from Malcolm's family, is dead. He recognizes that the people of Scotland have rallied behind Malcolm, and that they all agree he is the rightful next king for them.