Bukhari

Hadith Commentary:
The Messenger of Allah (sallallahu alayhi wa sallam) made the supplication:
O Allah! Grant honor to Islam through either Umar ibn al-Khattab or Amr ibn Hisham.
Allah, the Exalted, accepted the supplication of the Prophet (sallallahu alayhi wa sallam) in favor of Umar (radi Allahu anhu), and thus he embraced Islam.
(Jami‘ at-Tirmidhi, al-Manaqib, Hadith: 3681)

2.
What is meant by “they began to live a life of honor” is that after his acceptance of Islam, the Muslims began to pray openly at the Ka‘bah and the way for the propagation of Islam was opened.
Abdullah ibn Mas‘ud (radi Allahu anhu) states that the acceptance of Islam by Umar (radi Allahu anhu) was a source of honor for us, his migration was a help to Islam, and his rule was a mercy for the people.
By Allah! We could not pray openly at the House of Allah until Umar embraced Islam, and then we attained this blessing.
(al-Mu‘jam al-Kabir by at-Tabarani: 9/162, No. 8806)

In another narration, it is mentioned that when Umar (radi Allahu anhu) became Muslim, the polytheists said:
The Muslims have now taken revenge on us.
(al-Mu‘jam al-Kabir by at-Tabarani: 11/152, No. 11659)

In any case, in the early days of Islam, the disbelievers had the upper hand and would inflict great harm upon the Muslims, but when Umar (radi Allahu anhu) embraced Islam, the disbelievers ceased to harass the Muslims.
(Fath al-Bari: 7/61, 62)