In The Name of Allaah, The Most Merciful, The Bestower of Mercy.

Allaah [The Exalted] said:

الٓمٓ
غُلِبَتِ ٱلرُّومُ
فِىٓ أَدۡنَى ٱلۡأَرۡضِ وَهُم مِّنۢ بَعۡدِ غَلَبِهِمۡ سَيَغۡلِبُونَ
فِى بِضۡعِ سِنِينَ‌ۗ لِلَّهِ ٱلۡأَمۡرُ مِن قَبۡلُ وَمِنۢ بَعۡدُ‌ۚ وَيَوۡمَٮِٕذٍ۬ يَفۡرَحُ ٱلۡمُؤۡمِنُونَ
بِنَصۡرِ ٱللَّهِ‌ۚ يَنصُرُ مَن يَشَآءُ‌ۖ وَهُوَ ٱلۡعَزِيزُ ٱلرَّحِيمُ
وَعۡدَ ٱللَّهِ‌ۖ لَا يُخۡلِفُ ٱللَّهُ وَعۡدَهُ ۥ وَلَـٰكِنَّ أَڪۡثَرَ ٱلنَّاسِ لَا يَعۡلَمُونَ
يَعۡلَمُونَ ظَـٰهِرً۬ا مِّنَ ٱلۡحَيَوٰةِ ٱلدُّنۡيَا وَهُمۡ عَنِ ٱلۡأَخِرَةِ هُمۡ غَـٰفِلُونَ

Alif-Lam-Mim. [These letters are one of the miracles of the Quran, and none but Allah (Alone) knows their meanings]. The Romans have been defeated. In the nearer land (Syria, Iraq, Jordan, and Palestine), and they, after their defeat, will be victorious. Within three to nine years. The decision of the matter, before and after (these events) is only with Allah, (before the defeat of Romans by the Persians, and after, i.e. the defeat of the Persians by the Romans). And on that Day, the believers (i.e. Muslims) will rejoice (at the victory given by Allah to the Romans against the Persians), with the help of Allah, He helps whom He wills, and He is the All-Mighty, the Most Merciful. (It is) a Promise of Allah (i.e. Allah will give victory to the Romans against the Persians), and Allah fails not in His Promise, but most of humankind know not. They know only the outside appearance of the life of the world (i.e. the matters of their livelihood, like irrigating or sowing or reaping, etc.), and they are heedless of the Hereafter. [Surah Ar-Rum. Aayaat 1-7]

Imaam As-Sadi [may Allaah have mercy upon him] stated: In that era the Persians and Romans were among the most powerful states in the earth, and there used to be wars and fighting between them as happens between states of equal strength. The Persians were polytheists who worshipped fire, and the Romans were people of the Scripture who ascribed themselves to the Tawrat and Injeel, and they were closer to the Muslims than the Persians; therefore, the believers loved that the Romans should gain the upper hand and be victorious over the Persians, but the polytheists (of Makkah) – due to the religion of polytheism they had in common with the Persians – loved that the Persians should be victorious over the Romans. So, the Persians were victorious over the Romans, but the victory did not extend over their lands, rather it was limited to the areas closest to their frontiers. The polytheists of Makkah were happy and the Muslims were sad due to that, so Allaah informed and promised them that the Romans will overcome the Persians [فِى بِضۡعِ سِنِينَ‌ۗ – Within three to nine years] – Meaning, within nine or eight years, or approximately – neither exceeding ten years nor less than three years; and that the Persian victory against the Romans and the Roman victory against the Persians occurs by Allaah’s Will and decree. This why Allaah stated: [لِلَّهِ ٱلۡأَمۡرُ مِن قَبۡلُ وَمِنۢ بَعۡدُ‌ۚ – The decision of the matter, before and after (these events) is only with Allah]- Meaning, victory does not merely occur due to the means one employs, but rather it has to attached to Allaah’s Will and Decree.

[وَيَوۡمَٮِٕذٍ – And on that Day]- Meaning, the day that the Romans will overcome and defeat Persians, [يَفۡرَحُ ٱلۡمُؤۡمِنُونَ – the believers (i.e. Muslims) will rejoice; بِنَصۡرِ ٱللَّهِ‌ۚ – with the help of Allah; يَنصُرُ مَن يَشَآءُ‌ۖ – He (Allaah) helps whom He wills]- Meaning, they (i.e. the Muslims) will rejoice due to the Roman victory against the Persians, even though both parties are disbelievers, but some evils are less harmful than others, and on that day the polytheists (of Makkah) will be sad. (1)

Shaikhul Islaam Ibn Taymiyyah [may Allaah have mercy upon him] said:

The Persians were fire worshippers and the Romans were Christians. At first the Persians defeated the Christians and this occurred during the early days of the Prophet’s [peace and blessings of Allaah be upon him] mission whilst he was in Makkah and his followers were few; so the polytheists were happy due to the victory of the Persians, because they were closer to them than the people of the Scripture, and because both the Quraish and the Persians were nether people of the Scripture nor did they believe in the resurrection. The Muslims were sad because the people of the Scripture were closer to them [i.e. even though they distorted their books and are commanded to follow the final Messenger]. Then Abu Bakr As-Siddeeq [may Allaah be pleased with him] went to the Prophet [peace and blessings of Allaah be upon him] and informed him about the victory of the Persians, so Allaah revealed:

الٓمٓ
غُلِبَتِ ٱلرُّومُ
فِىٓ أَدۡنَى ٱلۡأَرۡضِ وَهُم مِّنۢ بَعۡدِ غَلَبِهِمۡ سَيَغۡلِبُونَ
فِى بِضۡعِ سِنِينَ‌ۗ لِلَّهِ ٱلۡأَمۡرُ مِن قَبۡلُ وَمِنۢ بَعۡدُ‌ۚ وَيَوۡمَٮِٕذٍ۬ يَفۡرَحُ ٱلۡمُؤۡمِنُونَ
بِنَصۡرِ ٱللَّهِ‌ۚ يَنصُرُ مَن يَشَآءُ‌ۖ وَهُوَ ٱلۡعَزِيزُ ٱلرَّحِيمُ

Alif-Laam-Meem. The Romans have been defeated. In the nearer land (Syria, Iraq, Jordan, and Palestine), and they, after their defeat, will be victorious. Within three to nine years. The decision of the matter, before and after (these events) is only with Allah, (before the defeat of Romans by the Persians, and after, i.e. the defeat of the Persians by the Romans). And on that Day, the believers (i.e. Muslims) will rejoice (at the victory given by Allah to the Romans against the Persians); with the help of Allah, He helps whom He wills, and He is the All-Mighty, the Most Merciful. [Surah Ar-Rum. Aayaat 1-5]

This was what the Prophet [peace and blessings of Allaah be upon him] informed them before its occurrence and it occurred as he stated. When Abubakr As-Siddeeq [may Allaah be pleased with him] informed the Quraish about it, they belied him, so he placed a price on that as mentioned by the scholars of Tafseer and the scholars of Hadeeth. Sufyaan Bin Sunayd [may Allaah have mercy upon him] – a Shaikh of Imaam Al-Bukhaari [may Allaah have mercy upon him] – stated in his Tafseer with a chain of narration to Niyaar Bin Mukram al-Aslamee [may Allaah be pleased with him] who said: When Allaah revealed these Aayaat to His Messenger [peace and blessings of Allaah be upon him], the leaders of Makkah said, ‘O Ibn Abee Quhaafah (i.e. Abu Bakr As-Siddeeq)! What is this that your companion [i.e. the Prophet] says?’ He said, ‘By Allaah, it is the speech of Allaah’. So they said, ‘Then let there be a bet between us on the basis that the Romans will defeat the Persians between three to nine years’; so Abu Bakr [may Allaah be pleased with him] betted them. Many polytheists entered into Islaam when that occurred’”. Ibn Mukram [i.e. Niyaar] said, “The Quraish wanted the Persians to be victorious since they were not people of the Book, nor did they believe in the Resurrection. So when Allah revealed these Aayaat, Abu Bakr As-Siddiq [may Allaah be pleased with him] went out, proclaiming throughout Makkah, ‘Alif, Laam, Meem. The Romans have been defeated. In the nearest land, and they, after their defeat, will be victorious, within three to nine years’. Some of the Quraish said, ‘Then this is (a bet) between us and you. Your companion claims that the Romans will defeat the Persians within three to nine years, so why not have a bet on that between us and you?’ Abu Bakr said, ‘This was before betting has been forbidden’. So Abu Bakr and the idolaters made a bet, and they said to Abu Bakr, ‘’What do you think – Bid’ means something between three and nine years, so let us agree on the middle.’ So they agreed on six years; then six years passed without the Romans being victorious. The idolaters took what they won in the bet from Abu Bakr. When the seventh year came and the Romans were finally victorious over the Persians, the Muslims rebuked Abu Bakr for agreeing to six years. He said, ‘Because Allah said: ‘In Bid’ years.’’’ Many people became Muslims at that time.” This hadeeth has also been reported by At-Tirmidhee Numbers, 3193 and 3194 and this story related to it is Mutawaatir. (2)

[Ref 1: An Excerpt from Tafseer As-Sadi. slightly paraphrased]
[Ref 2: An Excerpt from Al-Jawaabus Saheeh Liman Baddala Deenul Maseeh. Vol 1/ 84-91. slightly paraphrased]