As the other answers show, English is not the only language with this phrase, but we can examine the etymology of it to explain what it means.
When did this usage start?
The OED has
3d. A polite formula used in response to an expression of thanks,
with the first citation in 1907.
How did it originate? It is a natural extension of the use
3a. Freely permitted or allowed, cordially invited,
with the first citation from ca. 1400.
To see how 3d evolved from 3a, consider this exchange (from Google books, 1839)
“Do you really” – exclaimed Theophan, his voice trembling with joyful surprise – “do you really say so! – what shall I – how can I sufficiently thank you?”
“It matters not,” said the old man. “you are welcome to it.”
So we can see that "you're welcome" probably came from something akin to "it's nothing -- you're welcome to it". This changes its meaning from, "I'm acknowledging that I did you a favour" to a more generous sense of giving that's found in other responses like "it's nothing".
Why isn't this usage found in French (for example)?
I don't believe French has an adjective with the exact meaning of 3a. Looking up welcome in the Larousse English-French dictionary, they paraphrase all uses with meaning 3a. For example:
you're welcome to anything you need,
is translated as:
servez-vous si vous avez besoin de quelque chose
(help yourself if you have need of something).