First the numbers.
It seems hard to believe that millions of Muslims would be turning from Muhammed to Jesus. After all, most people don't change the religion they're born into. And Islam prescribes death for any Muslim who would convert to Christianity. So they don't do it lightly. But millions? Yes, according to researchers Duane Alexander Miller and Patrick Johnstone who published "Believers in Christ from a Muslim Background: A Global Census"  in the Interdisciplinary Journal of Research on Religion. Here is a graph from the article, and note this is only up to 2010.
Crosswalk.com  gives a more conservative number of converts in the range of 2 to 7 million, based on the work of missionary David Garrison, author of "A Wind In the House Of Islam" .
Since this appears to be a large number of converts, I want to explain more of Drs. Garrison, Miller, and Johnstone's methods. They all acknowledge that exact numbers are impossible, that numbers are both overreported and underreported, and even estimates are very difficult. But how do we get any estimates at all? If you wish to skip the data collection section, go to the section "How Is It Happening?".
In 1960, Dr Johnstone began compiling data on the size and nature of Christian communities worldwide. He published six volumes of Operation World , also see Wikipedia .
The purpose of the Operation World books has been to provide a profile of each country in the world, including information on the ethnic and religious breakdown, with the hope of aiding Christians and Christian ministries that focus on world missions, evangelism, and prayer. Religious and denominational information was gleaned from national surveys and censuses, solicited from missionaries and indigenous workers, and culled from official records of groups such as the Lutheran World Federation and the World Communion of Reformed Churches. IJRR 
The end result is a reference book of geography, demographics, population, literacy, economy, religions, Christian denominations, and missions groups for every country in the world. Here is their summary of Muslim converts to Christianity.
Garrison's approach focused solely on Muslims converting to Christianity. He conducted interviews with over 1000 converts in 14 countries using a questionnaire he developed. The house of Islam can be divided into nine geo-cultural rooms; three in Africa, the Arab world, the Persian world, Turkestan, West India, East India, and Indonesia. He went to all nine travelling 250,000 miles. He only counted movements with more than 1000 converts or more than 100 churches. The questionnaire itself can be found in his book "A Wind In the House Of Islam" . His questionnaire and interviews went beyond estimating numbers of converts, to an attempt to answer the questions of how and why the conversions were taking place, and why they are accelerating.
How Is It Happening?
During the first 13 centuries of Islam, the only conversions either way, Muslim to Christian or Christian to Muslim appear to have been forced. Christian missionaries had little to no success in proselytizing Muslims. In fact, missionaries were often forbidden from preaching to Muslims. The first breakthrough came from a local Javanese evangelist Radin Abas Sadrach Surapranata (1840 - 1924) remembered as "Sadrach, The Apostle of Java". His success was the product of at least two new factors - he was a local, and the recent publication of the Bible in Javanese, that is Muslims distrust Westerners, and now they could read the Bible in their native language.
Native language Bible translations were partly responsible for the success of William Carey  (1761 - 1834), aka father of modern missions, who translated parts or all of the Bible into 44 Bengali languages. Bengali is the sixth most common language spoken today. Again, people could see for themselves.
Native language Korans are also making a huge difference. Muslims can read the Koran and the Bible and decide for themselves.
Sheikh Bilal added, “The translation of the Qur’an is changing everything. Before, it was only in Arabic, which no one understood. - "A Wind In the House Of Islam" 
TheReligionOfPeace . But the Muslim to Christian converts experienced Islam up close and personal like none of us in the West, that is to say they have seen or felt the hate and cruelty themselves, indeed they have lived it. Here are some quotes from A Wind In the House Of Islam.
As we will see in many subsequent Rooms in the House of Islam, one of the greatest recurring motivations for Muslims coming to Christ is a rejection of the militant expression of Islam itself.
God is using dissatisfaction within the Muslim community itself to draw Muslims to faith in Christ.
[Berbers] saw in the way of Jesus a stark contrast to that of Muhammad.
Often enough, it is injustice within the Muslim community itself that leads Muslims to turn to Christ.
In these and similar testimonies, one observes a pattern. When the Muslim community mistreats one of their own, that person is more inclined to find an alternative sanctuary in the person of Christ and the Christian faith.
But Wait, There's More
DREAMS AND VISIONS: Is Jesus Awakening the Muslim World" - , "has encountered a staggering number of Muslims who were first introduced to Jesus through a vision or dream so powerful that they eventually turned from their lifelong religion of Islam and embraced Christ as their Savior". He also wrote "Killing Christians: Living the Faith Where It's Not Safe to Believe" - , which recounts the journey of several people coming to Jesus through dreams. Here is one exchange between a Muslim and a Christian.
I’ve been hearing a lot about Muslims having dreams about Jesus. And they’re not the kind of dreams someone has after a bad plate of baba ghanoush or something like that. These are high-definition dreams— visions!— of Jesus, and good-hearted Muslims who have them end up asking Jesus followers what they mean. Isn’t that cool?”Family members often have the same dream. But it doesn't end with dreams. They are driven to know more about the Jesus who appeared to them, and seek out Bibles. You can't base your faith on dreams and visions and ignore the written Word of God. As best I can tell, the dreams decrease in frequency when a person matures in their faith. They also find a group to worship with, maybe an established Western church, but often small home groups where they can meet in safety. And within weeks or months, these same people are out evangelizing in a war zone or refugee camp.
I personally have never had a dream or vision of Jesus. I have to wonder why He chooses such a direct method of calling Muslims. Then again, I've had access to the Bible in English all my life, and the peace and safety to attend any church I want. And visions aren't the goal, they are only an introduction, then a Bible (in their native language), then a church. Muslims turned Christian can be called to martyrdom, and they are willing to suffer death for Jesus. You might say Jesus gives them a lot because He's going to ask a lot of them.
But what about [insert your pet doctrine here]? No church has perfect doctrine. The Muslims turned Christians are showing their faith by their works. As one woman said "They're turning away from Islam!!!! God can fix the rest!" None of the dreams retold by Garrison or Doyle contradicted scripture. Not much is said about specific teachings other than who is Jesus, baptism as a sign of conversion, receiving the Holy Spirit, and evangelizing. What is clear is that lives are changing for the better. Muslim lives. Millions of them. The fields are truly ripe for harvest, pray that He sends laborers.
4. http://www.operationworld.org/6. https://en.wikipedia.org/wiki/William_Carey_(missionary)