The virtues of these ten days are based on many things:

Allah Swears an oath by the ten days of Dhul-Hijjah, and swearing an oath by something is indicative of its importance and great benefit.

Allah says:
وَٱلْفَجْرِ
وَلَيَالٍ عَشْرٍ
“By the dawn and by the 10 nights (i.e. the first 10 days of the month of Dhul-Hijjah)…” [Qur’an 89: 1-2]

Ibn ‘Abbas RA, Ibn Az-Zubayr RA, Mujaahid RA and others of the earlier and later generations, may Allah have mercy upon them, said that this refers to the first 10 days of Dhul-Hijjah. Ibn Katheer, may Allah have mercy upon him, said: “This is the correct opinion.” [Tafseer Ibn Katheer, 8/413]

The Prophetﷺ encouraged people to do righteous deeds because of the virtue of this season for people throughout the world, so even the non-Haaji gets to benefit from these days just by virtue of the days itself, obviously those undertaking the Hajj journey get the added benefit of the virtue of the place – the Holy Kaàba and the various sanctified places where the Hajj rituals get carried out.

The Prophet ﷺ commanded us to recite a lot of Tasbeeh (سبحان الله SubhaanAllah), Tahmeed (الحمد لله Al-hamdu Lillaah) and Takbeer (الله اكبر Allahu Akbar) during this time. ‘Abdullah Ibn ‘Umar, may Allah be pleased with him, reported that the Prophet ﷺ said:

“There are no days greater in the sight of Allah and in which righteous deeds are more beloved to Him than these 10 days, so during this time recite a great deal of Tahleel (“Laa ilaaha illAllah”), Takbeer and Tahmeed.” [Ahmad]

These 10 days include the Day of ‘Arafah, on which Allah perfected His Religion.

ٱلْيَوْمَ أَكْمَلْتُ لَكُمْ دِينَكُمْ وَأَتْمَمْتُ عَلَيْكُمْ نِعْمَتِى وَرَضِيتُ لَكُمُ ٱلْإِسْلَـٰمَ دِينًا
Today I have perfected your religion for you, completed my favour upon you and approved Al-Islam as a Deen (way of life for you). [Al Maida: 3]

Fasting on this day will expiate for the sins of two years. These days also include the Day of Sacrifice, the greatest day of the entire year and the greatest day of Haj, which combines acts of worship in a way unlike any other day.