Short answer
Because they are colouring grey or white hairs with henna, which turns white hair orange to red. The Prophet ﷺ told Muslims to change grey hair and avoid black, and praised henna and katam, so dyeing a grey beard with henna is mustahabb (recommended) in the Hanafi school and the other schools. Henna mixed with katam or indigo gives a darker brown, as long as the result is not black.
Dyeing grey hair and beards is mustahabb, and henna, alone or with katam, is the best dye according to the hadith. Pure black dye is makruh tahrimi for most purposes. Dyeing a beard that is not grey is permitted but is not this sunnah.
The four schools agree on this.
Explanation
When Abu Bakr's father, Abu Quhafa, was brought to the Prophet ﷺ with his hair and beard completely white, the Prophet said: change this with something, and avoid black. He also said that the Jews and Christians of his time did not dye, so be different from them, and that the best things to change grey hair with are henna and katam.
Why orange. Henna on dark hair gives only a reddish tint, but on white hair it turns bright orange or copper. That is why the men you see with orange beards are usually older men with grey or white hair following this Sunnah. Several Companions, including Abu Bakr, dyed with henna and katam.
The ruling. In the Hanafi school dyeing grey hair is mustahabb, and the four schools agree it is recommended. Henna mixed with katam (a plant dye) or indigo gives a deeper brown, which many prefer, as long as the mix does not come out black. Pure black is the one shade to avoid; see dyeing hair black. Other shades are allowed; see dyeing your hair.
Young men. If your beard is not grey, colouring it orange is not this Sunnah. It is a style choice: allowed, but not the recommended act the hadith describes.
Wudu and ghusl. Henna leaves a stain, not a layer, so once the paste is washed out your wudu and ghusl are valid. Plucking grey hairs instead of dyeing them is disliked; see plucking grey hair.
The evidence
Quran, Surah Ar-Room 30:54
۞ ٱللَّهُ ٱلَّذِى خَلَقَكُم مِّن ضَعْفٍۢ ثُمَّ جَعَلَ مِنۢ بَعْدِ ضَعْفٍۢ قُوَّةًۭ ثُمَّ جَعَلَ مِنۢ بَعْدِ قُوَّةٍۢ ضَعْفًۭا وَشَيْبَةًۭ ۚ يَخْلُقُ مَا يَشَآءُ ۖ وَهُوَ ٱلْعَلِيمُ ٱلْقَدِيرُ ﴿٥٤﴾
Allah is the one who created you from weakness, then made after weakness strength, then made after strength weakness and white hair. He creates what He wills, and He is the Knowing, the Competent.
Grey hair is part of the life Allah decrees, and the Sunnah teaches how to treat it.
Sahih Muslim 2102
Sahih... غيروا هذا بشيء واجتنبوا السواد ...
"Change this with something, and avoid black."
Said about Abu Quhafa's white hair: change grey hair, but not to black.
Sahih al-Bukhari 3462
Sahih... إن اليهود والنصارى لا يصبغون فخالفوهم ...
"The Jews and Christians do not dye, so be different from them."
Dyeing grey hair is a recommended distinction.
Sunan Abi Dawud 4205
Graded sahih... إن أحسن ما غير به هذا الشيب الحناء والكتم ...
"The best things with which to change grey hair are henna and katam."
Henna and katam are the preferred dyes.
Where the ruling is recorded
- al-Fatawa al-Hindiyya, Kitab al-Karahiyya, chapter on circumcision, hair and nails. Dyeing grey hair with henna is recommended for men and women; black dye is disliked.
- al-Durr al-Mukhtar with Radd al-Muhtar, Kitab al-Hazr wa al-Ibaha. On dyeing the beard and the dislike of black.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.