Well, Hinduism has over a billion followers worldwide, which makes it the third most popular religion in the world. And Shinto is practiced by about four million people mainly in Japan. There are also widely practiced folk religions through out Africa and China and other places some of which have multiple deities (which in total all of those folk religions adds up to hundreds of millions of people). So, it's not true to say multi-deity religions disappeared. They are still a pretty powerful force. And Buddhism has at least half a billion followers. It isn't a multi-deity religion but it isn't monotheistic either.

However, for why it became so common both Islam and Christianity proselytize heavily.

Rome controlled much of Western Europe and North Africa. When the Roman emperors converted they slowly made it inhospitable for non-Christians in the empire. At first, Christians just received special benefits and proximity. But, slowly, it became increasingly hostile to non-Christian religions and finally it was outlawed. It took a couple of centuries but, eventually, much of the Western empire converted. Not out of spontaneous love of Christ but because there wasn't a whole lot of choice.

Under Mohamed and after his death Islamic forces spread through out the middle east, North Africa, India and into Europe. Strongly encouraging the people they conquered to convert (although unlike what people think it was mostly not required but non-Muslims just had to pay a special tax).

After the fall of Rome in the West the Church took on a lot of power. They actively sought to convert non-Christian Germanic people through out Western Europe. At the same time the Church in the East (which eventually split with the West), also actively sought to convert non-Christian people in much of Eastern Europe and the Middle East. The Eastern Church had the benefit of having state endorsement of the Byzantine emperor. Which meant that a local tribe would convert to seek the approval of the emperor. Once Islam became a dominant force in the region there was a race between the two to convert a lot of the tribes in between the Caliphate and the Byzantine emperor. Some claim tribes would choose based on whether they preferred to drink (go Christian) or smoke hashish (go Islam), which is unlikely but illustrates the race. The Khazars, who were in the area, converted to Judaism to screw with both of them.

Once much of Europe was Christian, others converted in order to be able to do business with them or have marriage alliances. For instance, once the Norse lands became separate, unified kingdoms they quickly converted, at least officially. Because they had to be able to maintain alliances with other kingdoms (and the help of a Church bureaucracy). The Catholic church was an important part of the feudal structure in Medieval Europe. And people converted to be close to power. There were also "crusades" against polytheistic tribes in Eastern Europe, which were brutal and not always successful. These were mostly taken up by the Germans who personally converted much of the territory east of them.

Long story short, they won because they fought hard, sometimes brutally so, to convert a lot of people. Most polytheistic religions were not ones that actively sought conversion of others. Buddhism also actively recruited new adherents and survived.