So how did potatoes come to their present popularity? The generally accepted story is that a French army officer named Parmentier was taken prisoner during the Seven Years War (1756-1763), and ate potatoes as part of his prison diet in Hamburg, Germany. He found that he liked them. After his release, he managed to introduce them to the French court ("Your majesty, the potato. Potato, I have the honour to introduce King Louis XVI and Queen Marie Antoinette. Introductions all the way round.") Marie Antoinette reportedly once wore a potato flower as a corsage. But she decided to take a break from eating cake and so ate potatoes. What the queen did was what everyone did, so the potato became fashionable and entered French cuisine. From France, to the world.
And so we arrive at your question. For also in the 1840s, pomme frites ("fried potatoes") first appeared in Paris. Sadly, we don't know the name of the ingenious chef who first sliced the potato into long slender pieces and fried them. But they were immediately popular, and were sold on the streets of Paris by push-cart vendors.
Frites spread to America where they were called French fried potatoes. You asked how they got their name--pretty obvious, I'd say: they came from France, and they were fried potatoes, so they were called "French fried potatoes." The name was shortened to "french fries" in the 1930s.
By the way, the verb "to french" in cooking has come to mean to cut in long, slender strips, and some people insist that "french fries" come from that term. However, the French fried potato was known since the middle 1800s, while the OED cites the first use of the verb "to french" around 1895, so it appears pretty convincing that "french fried potatoes" came before the verb "frenching." The origin of the name is thus the country of origin French and not the cooking term french.
French fries are commonly eaten with ketchup in the U.S., but with malt vinegar (delicious) in the U.K., and with mayonnaise (appalling) in the Netherlands. The French mostly take them straight, but the Belgians have the best idea (as is so often the case with food): they eat frites with buckets full of mussels.