Principle number 20 of Basic Elements of Islam: Non-Muslims must pay a large tax. Once a country is following Sharia law, non-Muslims are given the choice between becoming Muslim or becoming a dhimmi. Dhimmis are allowed to practice their non-Muslim religion if they pay the jizya (a tax). If they convert to Islam, they no longer have to pay the jizya. This obviously creates a practical incentive to convert. Read more.
This is ingenious. The tax takes money away from non-Muslims and their competing religions and gives that money to support Islam. The income from these taxes (usually a 25% income tax) helped fund the Islamic conquests during the first two major jihads. They conquered vast lands, most of them already filled with Christians and Jews, many of whom did not convert at first, and their jizya poured huge sums of money into the Islamic war machine.
Eventually, the numbers of Christians and Jews in those countries dwindled down as they converted or escaped (or in some cases, were massacred), until now, in most Islamic countries, Jews and Christians are very small minorities.
The tax-the-non-Muslims idea helps the Islamic idea-collection make more copies of itself by suppressing competing religions and financially supporting Islam.
Several ideas within Sharia law extend this effect. For example, non-Muslims are not allowed to build any new houses of worship. They're not even allowed to repair already-existing churches or synagogues. This puts the houses of worship of any competing religion in a state of permanent decline.
Also, non-Islamic prayers cannot be spoken within earshot of a Muslim — again, preventing Muslims from being infected by a competing religion. No public displays of any symbols of another faith may be shown either.
All of this prevents the spread of any competing ideas, and makes competing religions die out over time. That's why today there are so many "Muslim countries." Almost every other country in the world is made up of different religions, but because of these principles, Islam tends to displace all other beliefs and cultures wherever it becomes established.