Oh Lord, they're going to consume us all!
Anyway, to make a few points:
1) The video treats Islam as if it is inherently bad and violent. That it is not. Islam, like all major religions has an inherently good message, but of course there are some whack-jobs who pervert it to fit their own needs, as in all religions. The video also fails to differentiate between the differences between extreme Middle Eastern Islam and the brand of Islam practiced just about everywhere else.
It can safely be assumed that only a fairly small percentage of Middle Eastern Muslims are the extreme fundamentalists that do hold the scary viewpoints. Then, take into account that the Middle East is only the 3rd-most-populous Muslim region on the planet (behind Indonesia and Africa). The video implies (through ignoring population statistics) that all Muslims hold extreme beliefs and are aiming to take over the world which, when any amount of critical thought is applied to it, is just stupid.
2) It ignores the historical presence of Muslims in Europe. Islam was born in the Arabian Peninsula in the 7th century A.D., and by 711 A.D., the Muslims had conquered the Iberian Peninsula, which they held for over 700 years. By the time the various Caliphates had morphed into the Ottoman Empire, Asia Minor was entirely under Muslim control, and they soon pushed through into the Balkan Peninsula, capturing what is now Greece, Albania, Bulgaria, parts of Romania, the former Yugoslav states, the Caucasus, and parts of Ukraine and Russia, leaving large Muslim populations in most of those countries.
Islam didn't have much staying power in Spain, thanks to the Spanish Inquisition, but it has always had a very strong presence in the Balkans and in parts of Russia. Furthermore, these Muslims (much like the ones in Indonesia) are MODERATE. They are, like most European Christians, very secular in their everyday activities. What is there to make one believe that any new influx of Muslims are going to be the crazy ones?
3. The video failed to differentiate between "Islamic Republic" and "Republic with a majority population of Muslims." For example, Iran, a theocracy, is an Islamic Republic. Turkey, on the other hand, has a secular government, but a majority of the population is Muslim. Simply having 50%+1 of your population be a certain religion does not make your country a "______ Republic." If that were true, the US would be a Christian Republic.
4. It assumes all the Muslims that move to Europe are the crazies and also assumes that the very secularized culture of Europe will not have any tempering effects on the crazy ones. By immigrating to a new land, people are often forced to pick up on certain aspects of the culture, and from what I know about Europe (I'll admit I've never been), it is very secular and non-religious. One would think that this type of atmosphere would "rub off" on some of the more fundamentalist ones and would only encourage ideological moderation, much like the Balkan Muslims.
Just because Muslims are a very fast-growing population doesn't mean the world is going to hell in a handbasket. Various world governments are doing that fine without the help of religion.