"Lies onto" is not used in English in this sense. It's just "lies on," or if you need to be more clear about the motion, then "lies down on."
Yes, I've been giving this kind of feedback for weeks with regard to English prepositions. It hasn't helped.
To be fair, prepositions are always one of the hardest parts of learning a new language, I think - compare French, where there are two words for "in", dans and en, which aren't interchangeable (dans = physically inside of, en = everything that wouldn't fit under dans) and which people constantly mix up.
But yeah, the near constantly wrong prepositions in English sentences doesn't say much for the course creators' fluency...
I agree that it's frustrating. But perhaps the creators allow themselves to use unorthodox English to emphasize the subtle differences between e.g. újságon and újságra, which cannot easily be rendered in English in any less awkward way.
Does this mean the cat is moving onto the newspaper to lie down? If I say, a macska az újságon fekszik, this would simply the cat is lying on the newspaper, with no mention of motion?
Yes, exactly! Using "újságra" here means that the cat is in the act of lying down, rather than already lying there.
Also, the audio doesn't sound quite right... like the speaker is saying "le" instead of "ra".
"lies" = is in a horizontal position
"lies down" = moves into a horizontal position