This is largely not true.

The question of how Islam spread after the time fo the Prophet (S) is complex because there were many factors involved.

It is also complicated by the fact that the early Arab-Muslim empire (the caliphates) did spread by conquest. However, in general, they were focused on conquering land and did not force people to convert to Islam. Sometimes, they even discouraged it because non-Muslims would pay the jizyah tax. Also, in some remote regions, the presence of the caliphate was really in name only.

On the other hand, it can be said that having a ruling elite who was Muslim created social conditions that were favorable for people living in this empire to become Muslim because Islam was seen as the religion of the educated and powerful and the cultural elite. (How times have changed!) Still, when you look at the well-known institutions in Baghdad, Andalusia, etc, there is clear evidence of Muslims and people of other faiths working together or interacting.

In any case, after the Arab-Muslim conquests, conversion was not instant, and it took a few hundred years for North Africa and the Middle East to become a majority Muslim region. A similar thing happened later with the expansion of the Ottoman Empire into parts of Eastern Europe.

Of course, empires were common in the pre-modern era, and the Arab-Muslim empire was not different structurally from other empires which also had their own religious tendencies. 

Beyond that, it seems that people's reasons for converting to Islam in the past were varied and not dissimilar to the reasons that people have today. For instance, some people favoured the Islamic view of monotheism as simpler than that of other faiths. Some people liked the idea of social equity that, in theory, Islam promises. Some people were affected by the Qur'an. Some people converted due to marriage or for other social reasons, and so on. 

The spread of Islam outside of the early Arab-Muslim Empire (such as into India, China, Indonesia, and sub-Saharan Africa) occurred largely through contact with Muslim traders and Sufis as well as exiles, expatriates, etc. This also occurred over time and was not instantaneous.

I am not saying that no one was ever forced to convert, since in over a thousand years of history and along vast geographical regions, virtually anything and everything that can happen does happen, but just that, in general, there is no evidence that the masses were forced to convert to Islam at the point of a sword. 

Unfortunately this has become a very politicized subject today, both in the West and in some other regions, and there are a lot of falsehoods circulated to try to inflame sentiments against Muslims in areas of sectarian tension. I hope and pray that as human beings we can resolve these differences and move forward in more peace!