I do not think they even started actively advertising until 20 - 25 or so years ago and they are not very aggressive advertisers even now. Some how it just caught on and traveled the world.
Sometimes I wonder if people from places other than Louisiana like Tabasco more than people from here because I don't really like Tabasco and neither do a lot of people I know who otherwise like hot sauce. I love everything Tabasco makes other than their main hot sauce. Their garlic hot sauce is one of the best hot sauces I've ever had in my life. Their pepper jellies are amazing, but I just never liked their hot sauce that much. I don't hate it and am very happy to have it rather than nothing, though.
As for how they spread so far and wide, I think I've heard (not sure if this is true) that they gave little bottles to soldiers in WWII. Also, their bottle and label are sheer perfection. Everything about their packaging is brilliant -- the shape of the bottle, the diamond shaped label, the red top, the green neck label...it really is perfect packaging.