Could you please explain the meaning of the Hadeeth (narration) in which the Prophet, sallallaahu ‘alayhi wa sallam, ordered the head cut off of whoever wants to disunite Muslims during the time of temptations and trials?

All perfect praise be to Allaah, the Lord of the worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.

In the Hadeeth that you have referred to, the Prophet, sallallaahu ‘alayhi wa sallam, told us about some of the afflictions that Muslims would experience, advising them to stay united. In order to achieving this purpose, Muslims have to stop anyone who wants to sow discord, warding him off even by killing him, unless he stops these attempts peacefully. This is because the Muslim community's welfare and unity are more significant than killing a dispute-monger, even if he is a Muslim, according to the explanation of An-Nawawi . 

And Allaah Knows best.