The verse of the veil was revealed because Muhammad’s perverted companion Umar had a habit of spying on women when they left their homes to relieve themselves. The veil would thus help him differentiate between Muslim women from non-Muslim women or slave women.

Narrated Abd Al-A’laa from Mo’ammar from Al-Zuhri from Anas who said:
Umar saw a slave-girl wearing Hijab then he beat her and said: “Do not make yourself like free women!”
A slave girl of Muhajirun or Ansaar came to Umar wearing Jilbab (complete Hijab), he said: “Have you been freed?”
She said: “No!”
He said: “Put it off your head.”
Jilbab is for the freed women.
So she hesitated, so he got up to her with the whip (darrah), and he hit her on the head, until she threw it.
Al-Musannaf, Volume 3, Page #128
Narrated Aisha:
The wives of the Prophet used to go to Al-Manasi, a vast open place (near Baqiat Medina) to answer the call of nature at night.
Umar used to say to the Prophet: “Let your wives be veiled,” but Allah’s Apostle did not do so.
One night Sauda bint Zama, the wife of the Prophet, went out at Isha time and she was a tall lady.
Umar addressed her and said, “I have recognized you, O Sauda.”
He said so, as he desired eagerly that the verses of Al-Hijab (the observing of veils by the Muslim women) may be revealed.
So Allah revealed the verses of “Al-Hijab” (A complete body cover excluding the eyes).
Sahih al-Bukhari 146
Narrated Aisha:
Umar bin Al-Khattab used to say to Allah’s Messenger: “Let your wives be veiled”.
But he did not do so.
The wives of the Prophet used to go out to answer the call of nature at night only at Al-Manasi. Once Sauda, the daughter of Zama went out and she was a tall woman.
Umar bin Al-Khattab saw her while he was in a gathering, and said, “I have recognized you, O Sauda!”
He (Umar) said so as he was anxious for some Divine orders regarding the veil (the veiling of women.)
So Allah revealed the Verse of veiling. (Al-Hijab; a complete body cover excluding the eyes).
Sahih al-Bukhari 6240
Aisha reported that Sauda (Allah be pleased with her) went out (in the fields) in order to answer the call of nature even after the time when veil had been prescribed for women. She had been a bulky lady, significant in height amongst the women, and she could not conceal herself from him who had known her.
Umar b. Khattab saw her and said: “Sauda, by Allah, you cannot conceal from us. Therefore, be careful when you go out.”
She (Aisha) said: She turned back. Allah’s Messenger (may peace be upon him) was at that time in my house having his evening meal and there was a bone in his hand.
She (Sauda) cline and said: Allah’s Messenger. I went out and Umar said to me so and so.
She (Aisha) reported: There came the revelation to him and then it was over; the bone was then in his hand and he had not thrown it and he said: “Permission has been granted to you that you may go out for your needs.”
Sahih Muslim 26:5395

Hijab rules were introduced to distinguish free Muslim women from non-Muslim women and slaves (who could be freely abused):
O Prophet, tell your wives and your daughters and the women of the believers to bring down over themselves of their outer garments. That is more suitable that they will be known and not be abused. And ever is Allah Forgiving and Merciful.
Koran 33:59
For more, see: Koran 33:59.
Thanks to Hatun Tash and commentators for video and sources:
Muhammad “revealed” Koran 33:59 in response to Umar’s wishes. Thus showing Allah is Muhammad and Muhammad is Allah, so Muhammad was a false prophet, for a false god, Allah, for a false religion, Islam, without even a false book, Koran.
