As in all things, it's complicated. Speed of growth is measured as the annual growth rate, which is expressed as a percentage.
At a global level, Islam is growing faster than Christianity: for the past several years Status of Global Mission (most recent one is always on my Resources page) reports Islam is growing globally at about 1.87% p.a., while Christianity's is growing globally at a rate of 1.3% and the population's global AGR of 1.1%. Both Christianity and Islam are growing faster than the world population (so the world is becoming more religious); Islam is growing faster than Christianity, so the % Muslim in the world is increasing.
However, the "fastest" AGR doesn't equate to "the biggest religion." Total Christians are significantly larger than total Muslims - 2.4 billion vs. 1.7 billion - and even by 2050, there will still be over a billion more Christians (3.4 billion) than Muslims (2.6 billion).
Further, while Islam has a higher growth rate globally, Christianity has a higher growth rate in most individual countries. Islam's growth rate is higher in 93 countries. Christianity's growth rate is higher in 137 countries. (I am here comparing the AGR as given in the last edition of Operation World). Most of Islam's increase comes from demographic growth (births) in high-birth-rate countries where Islam is the majority. In countries where Christianity is the majority, the birth rate is lower (note--correlation, not causation). In most places, the twin benefit of demographics and conversion gives Christianity the edge. (Very little of Islam's growth rate comes from any kind of conversion: what conversion happens is nearly always in the context of marriage.)
A religion's growth rate tends to 'outstrip' the competing religion when the 'winner' is small. Smaller groups grow faster. In 8 of the 10 countries where Islam has the highest % of the country, Christianity has the highest AGR. However, in all of these ten countries (Sahara, Yemen, Morocco, Afghanistan, Mauritania, Somalia, Tunisia, Maldives, Comoros, Iran), the Christian population is a very tiny minority, and even with existing rapid growth rates it will not overtake Islam's population any time soon.
Islam has the advantage in places where the birth rate is high, and some level of conflict or pressure leads to emigration of other religions. Examples include Niger, Djibouti, Turkey, Jordan, Turkmenistan, Iraq, Pakistan, Tajikistan, Kyrgyzstan, Qatar, Azerbaijan, Mali, Egypt, Uzbekistan.
In the largest populations of the world, it's a mixed bag:
So when someone says Islam is the fastest religion growing in the world, the answer is "Yes, but..." Don't leap to the conclusion that Islam is the winner in terms of adherents. Christianity is larger in numbers globally, growing fastest where Islam is strongest, and certainly "in competition" in the largest populations.
Further discussion: Who is growing fastest: Islam, Christianity or Evangelicals. Map: Where the church is growing fastest.