Why Sunnis are more widespread comes primarily from the Rashidun Caliphate and the split of Sunni and Shia from the start. The split came over determining whether Abu Bakr or Ali should have been the successor of Muhammad. Eventually Ali did become the Caliph, but Shia recognize him as the first rightful Caliph. Well, there was about 24 years of Rashidun conquering, expansion, proselytization, and history before Ali became Caliph. So the idea that the rulers that were doing this conquering, expanding, proselytization, and such is insulting to Sunni and doesn’t really catch on with most new converts.

Over that 24 year period the people who recognized Abu Bakr as the first Caliph were simply the majority. They were the ones in power and in control, so as the Caliphate grew and Islam spread the Sunni recognition of Abu Bakr spread. Ali became Caliph when his successor predecessor was assassinated and from there was the First Fitna, Fitnat Maqtal Uthman, which saw the overthrow of the Rashidun Caliphate and it was replaced with the Umayyad Caliphate. This saw Aisha, Mummad’s wife, leading an army against Ali.

Sufis are primarily Sunni as well. Sufism is a perspective of mysticism within Sunni Islam, more or less characterized by rituals. A more “spiritual” activity of Sunni Islam, in the New Age American sense of “spirituality”. Sufism spread into Shia Islam after the Medieval period. Sufism is like Islamic Asceticism, if that makes enough sense. The Christians of the Medieval period practiced this kind of Asceticism as well, hence ritualistic self-flagellation etc. Sufi Sunni’s could be compared to Catholic Christians in the way that they ritualize their worship, recite prayers, and have doctrinal rulesets for how they do things.

So, Sunni Islam spread because Islam Spread. It was just the way things were. Sunni Islam was Islam. It wasn’t until 680 that Shia Islam really appeared and had an identity and that was 50 years after the death of Muhammad. That isn’t to say that it didn’t spread at all, a Shia state was possibly the first Moroccan state but that wasn’t established until another 100 years later. They had states periodically over Maghreb and the Levant. Eventually the entire Mediterranean coast from Arabia to Tunisia was ruled by the Shia Fatimids in about 900. The Fatima had a pretty great deal of religious tolerance and did not successfully convert too much of their population. By 1200 that Caliphate was being replaced/conquered by Sunni rulers from both East and West. The only places Shia maintained their majority rule was in modern Lebanon and Iran. They have been the minority almost everywhere else.

Someone else, u/Velostodon, already replied to this question, so I’m going to formulate my answer in a way that, in a way, is more of an answer to his. Although his answer is very consise and to-the-point, I feel that it overlooks many of the very interesting developments in early Islam as well as in early medieval Shiʿism, and I disagree strongly with the notion that Sunni Islam must be considered the “natural” form and Shiʿa Islam the “exception”. If you feel that I mischaracterize you, then I apologize. Let’s get started:

 

I - The catalyst: martyrdom of the forefathers

u/Velostodon starts by referring to the Rashidun caliphate, which consisted of four elected rulers (Abu Bakr, ʿUmar, ʿUthmān and ʿAlī) and mentions the split between the Sunnis and Shiʿa’s from the start, stating that the conflict arose due to a disagreement whether ʿAli or Abū Bakr should have been the first Caliph. This is mostly accurate: although there was a sizeable faction in the early community that argued that ʿAli should be the Prophet’s successor, the majority voted for Abu Bakr. In Shiʿi historiography it is often mentioned that the decision was rushed through while ʿAlī was attending to Muḥammad’s corpse, but the writings detailing this mention from the second century after the fact, so it’s impossible to establish if this actually happened. Either way, it is said that ʿAlī did not contest Abu Bakr’s election.

 

What is important to note, however, is that the majority of ʿAlī’s supporters came from the so-called ʾanṣār (‘the helpers’) and some of the very earliest muhağirūn who had come with Muḥammad to Madīna: they represented the kind of Muslim with unimportant or no tribal affiliation (such as ex-slaves) or Arabs with little social standing. Abū Bakr’s claim was maintained by the establishad tribes of Mecca, some of whom had until very recently opposed Muḥammad, even violently. The ʾanṣār, perhaps justifiably, feared that the Meccan aristocracy would turn the situation in their favour. However, what is important to note is that at this point, the situation was still wholly political.

 

The situation only started to gain a spiritual dimension when ʿAlī rose to power: his predecessor, ʿUtmān, was assassinated by a group of disgruntled Egyptians. Due to his considerable tribal affiliations, his cousin Muʿāwiya insisted that ʿAlī would execute his assassins. However, ʿAlī was already in a politically weak position, as the majority of his support came from Egypt, as well as from Southern Iraq. His refusal to do so led to open rebellion on Muʿāwiya’s part, who was already governor of Syria at the time, and who gained support from Muḥammad’s famous widow, Aisha. At this time, most historians prefer to use the term "ʿAlids" or "proto-Shiʿites", as they were a relatively diverse group with no particularly established orthopraxy.

 

After a short skirmish in Iraq, Muʿāwiya and ʿAlī declared a truce. At this point, a sizeable portion of ʿAlī’s supporters left him, arguing that Muʿāwiya’s rebellion had turned him into an non-believer (kāfir), as ʿAlī had been divinely appointed. ʿAlī’s compromise with a non-believer meant that he lost his divine mandate. This group was henceforth known as the ḫāriğiyya (‘those who went out’), and would continue to wreack havoc on the Islamic world for the next two centuries. The rest of ʿAlī’s life was rather miserable: he was no longer in control of some of the richest areas of the Caliphate and his legitimacy amongst his followers was severely diminished. For the next six years he lived in Kūfā, until one of his ex-followers stabbed him with a poisoned dagger as he was praying, dying two days later.

 

The Caliphate was now firmly in control of Muʿāwiya, who was able to rule in relative peace for a decade or so, although his governance was never really accepted by ʿAlī’s earlier supporters. He was able to bribe al-Ḥasan, who was ʿAlī’s oldest son, from rebelling against him; although the Shiʿi sources prefer to mention that Ḥusayn was being a pacifist. Two years before Muʿāwiya’s own demise however, al-Ḥasan died suddenly in Mecca (of course, the Shiʿi sources blame Muʿāwiya for this). Ḥusayn, who was al-Ḥasan’s brother and successor, was not as content with Muʿāwiya’s governance, and there is a famous letter transmitted by Shiʿi historians that describe some of their interactions (al-Tusī; Kitāb al-Iqtirāḥ (c. 1150)):

 

I know not of any trial (fitna) to the Muslim community greater than your governance; not in the eyes of my soul, nor my faith, nor in that of Muḥammad’s community; nor is anything better for me than to strive against you. Doing this would surely be pleasing in the eyes of God. [...] Know, oh Muʿāwiya, that God (the most Exalted) in his Reckoning does not ignore the great nor the small, He merely counts them. God’s supporters will make war unto you, and they will cast you from your house into exile; and those people who support your son, a drunk slaveboy who plays with dogs, will know nothing of you, except that you have lost your soul, destroyed your faith, misled your people, and ruined your trustworthiness by listening to the words of insolent ignoramuses, having killed your piety on their account.

 

As you can see from Ḥusayn’s letter, Muʿāwiya’s wrongdoing were considered both religious and secular; The pot boiled over when Muʿāwiya appointed his son Yazīd as successor, the first time a Muslim ruler had attempted to create dynastic rule, and when Muʿāwiya passed away, it finally led to another open revolt. Ḥusayn was invited to come to Iraq, and having set out from Madīna, he was set upon by one of Yazīd’s armies and was killed in the following battle near Karbalā, Iraq. This event is extremely important for Shiʿites and Ḥusayn’s death is commemorated every year on the 10th of Muḥarram.
(cont.)

The answer is very big and very complicated, and people have written entire dissertations on tiny subsets of what I'm about to write. I'm going to try and stay away from the comparing the qualities of the three sects and focus more on the historical aspects of the spread of Islam.

Right off the bat, Shia and Sufi Islam didn't stay confined to the middle east, you can find small communities of Shiites and Sufis everywhere you go. A better question then is this: why is Sunni Islam the majority faith outside of the Middle East? More importantly, why is Sunni Islam the majority sect in Islam?

That last question has already been answered elsewhere in the thread, so I'll move onto the reasons why Sunni Islam is generally more prevalent. What has also been mentioned is that Sufism is less a religion and more of a mystical approach that one can take towards Islam. What hasn't been mentioned is that Sufi orders were a huge factor in the spread of Islam. Sufis would establish orders, convert nearby towns, and move onwards and outwards to newer, untamed pastures. However, despite their widespread popularity, many of these orders were eventually disbanded by the Ottomans, who viewed them as a threat to their centralized and fairly secular power. So, the former ottoman territories are Sunni.

The eastward spread of Islam into the Indian subcontinent is due the conquest-oriented efforts of the Ghaznavids, Ghurids, and finally Mamluks, who all practiced Sunni Islam. Thus, that area was Sunni at first. While the Islam practiced there later turned into something entirely unique (which is a fascinating subject, look into Akbar the Great), it was always rooted in Sunni philosophies and ideas. So the Indian subcontinent is Sunni.

Malaysia and Indonesia became Muslim in what one would call the "traditionally Islamic" way: traders practicing the religion spread their faith by virtue of practicing it well. They came from Muslim India and Arabia, bringing Sunni Islam with them. So southeast Asia is Sunni.

I confess I know very little about Sub-Saharan Africas relationship with Islam. What I can tell you is that the spread of Islam there was frequently in the traditional Sufi way. So Sub-Saharan Africa is Sufi/Sunni.

Europe has always been in contact with the Sunni dynasties of the Middle East, especially after the the conquest of Spain and much, much later the fall of Byzantium. In America, however, there's been a roughly even distribution of Sunni and Shia immigrants. The first mainstream exposure to Muslims was through the Nation of Islam, who practiced a form of Sunnism. More importantly though, since the 80s, Saudi Arabia has been financing missionary efforts all throughout the West. Since Saudi Arabia is sunni, most western converts are as well.

This is a heavily, heavily simplified account of what happened to cause Sunni Islam to be in the majority world wide. If you have any specific questions about a region or time period I'd be happy to answer to the best of my ability.