There is a litany of sources for the popularity of the tricolor in Western nations. First there is the concept of representation of the people, or at least a nationalist movement, rather than a monarchy or religon. Second there is the phenomenon of “flag families” whence flags derive traits and styles from the influence of other nations’ flags.

The tricolors in the West typically stand to represent the people, a nationalist movement, or otherwise the nation itself, at least in a political context. You’ll notice that in many cases the flags which most tricolors replaced were of royal origin (take the Bourbon fleur-de-lis replaced by the French tricolor). The heraldry on many non-tricolor European flags remains monarchial or religious in some regard, and allude not to the people or nation itself but to other important symbols of the nation, be it a monarch (Spain, Portugal, San Marino, even the allegorical tale associated with the Austrian flag to a lesser extent), or religious symbols (Scandinavian countries, Vatican, etc.). Now that is not to diminish the developed connotations these flags have for their people, and what they have come to represent, merely to point out the origins.

The concept of “flag families” is an interesting one in the vexillological community, whereby some flags may credit their attributes to other countries. One of the principle cases being the Italian tricolor taking after France. In this manner, some countries have shown their relative stance and political situation in their flag, with Belgium adopting a vertical tricolor in deference to France rather than the Netherlands. This has had a converse effect on many nation’s flags as the horizontal stripe also demonstrated national sentiments without the connotations of bloody revolution that came with the French tricolor. The horizontal stripe in turn has come to represent many nations in the same manner as the vertical tricolor, displaying important national ideals and replacing religious and heraldic imagery. However each flag really is a case-by-case basis.

Ultimately each flag has its own story, but the tricolor brings a new beginning to the nation, as it is essentially an arbitrary symbol chosen to represent the nation without the connotations that former symbols may have had in the nation.

TLDR - Tricolors evolve from national movements and normally replace religious or heraldic symbols, or they may copy the trends of other flags.

Maybe try /r/vexillology?

Irelands is representative of our two major (religious) groups and the peace between them. Our flag is green, white and gold (orange). The green represents the Catholic Irish. The gold/orange represents the Protestant Irish. And of course the white in between represents the peace between the two.

Edit: Sources in child comment below.

A lot are imitative of the Dutch Republic flag. It started out with horizontal Orange-White-Blue stripes but the orange dye would fade into red on sailing ships. So the Dutch, being Dutch, just decided to go with red.

Not too long after that Peter the Great of Russia took his tour of western Europe to try and discern what made them so advanced. He was taken by the way the Dutch ran things, particularly their shipbuilding industry. So he imported a bunch of Dutchmen to help build a Russian navy. He was so taken he modeled a flag for Russian ships based off of the now Red-White-Blue Dutch flag. This White-Red-Blue flag was eventually adopted for all of Russia.

Fast forward to the 19th century as various Slavic nationalities were breaking away from the Austro-Hungarian and Ottoman Empires. The last time these guys were independent was when flags were for dynasties, not nations. Due to the influence of the Pan-Slavic movement that was heavily promoted by Russia, a whole mess of these countries based their flags off of the White-Red-Blue that the Russians originally copied from the Dutch. In addition, a bunch of non-Slav Russian Empire successor states chose to stick with the horizontal bars and different colors from the Slavic red white and blue.

While we are at it, the dynastic symbol of the Ottomans was the star and crescent. It has no meaning in Islam, its just that for centuries Christianity's contact with the Islamic world was mostly through the Ottomans, so the star and crescent sort of become synonymous with Islam. Anyways, Muslim countries that were broken off the Ottoman Empire before the era of defined Arab Nationalism (WWI era) tended to keep the star and crescent in their flags. Countries broken off after WWI adopted the Arab Nationalist pattern cooked up by the British and Arabs in WWI, specifically to get rid of the traditional star and crescent. The Bosnians made an attempt to use the star and crescent after the fall of Yugoslavia.

So there you have it: most of the flags in Europe with horizontal tricolors are influenced by the Dutch flag. It would seem the Irish, Italian, and Belgian flags are influenced by proximity to France or sympathies with the revolutionary republican model. And to round out the Mediterranean, Turkey and most North African flags are inspired by the Ottoman dynastic symbol. Arab flags are modeled on European patterns.

Follow up to this, why did all Scandinavian/Nordic countries adopt the same left-oriented cross style flag?