(Now I don't know where Estonian and Finnish came from, but I do know a bit about the history of Hungary, as it often ties in with our own (= Czech) history.)
The Hungarians were actually a people that lived in a very "nomadic" style. They came to the Central Europe somewhere from the East (I think that the Hungarian is considered an "Uralic" language, so maybe they came from Ural?) at the beginning of the 10th century and they started raiding everything they could (and since they were essentially living and fighting on the horseback, they were quite good at it). By the way, one of the first things they did is that they made a quick end to the Great Moravia (that was in ~907). Maybe there were other causes too, but we don't know much about it.
After that, they repeatedly wandered through today's Austria, eastern parts of Germany, and of course today's Czech Republic and Slovakia, pillaging and taking what they could. On some occasions they were repelled (and forced to take a break with the pillaging) but all in all, they were pestering the Central Europe for nearly 50 years, until they have finally been given a sound thrashing in 955 at the river Lech (essentially the Germans and the Bohemians alike were so much fed up with them that they joined forces and defeated them. There were more battles than this one, but we don't know much about them.).
When much of their forces had been defeated, the rest of them was allowed to settle peacefully roughly in the places where today's Hungary is (sort of pushing/mixing with the original Slavic population), and they rose in power very quickly, so at the break of the millenium they already had things like a hereditary title of a king, and a seat of an archbishop (while we the Bohemians managed to get those boons much much later). All in all, a new power was in place. For much of the history, they were spanning much farther than today's Hungary (for instance, they held much of today's Slovakia, which was called "the Upper Hungary"), they were reduced in size only after the First World War, and even before and in the Second World War, the Hungary sided with Hitler because there was a widespread belief that with him, they will be able to recover their country to its former glory of "Great Hungary". (Essentially "Make Hungary great again", if you permit.)
I think that now it's clear: the language is non-Indo-European because it was spoken by people who came from far away and settled only later in the middle of the otherwise Indo-European nations.