To answer your first question: it is NOT very close at all.
TFA Visual dictionary includes a map of the Galaxy, and Takodana looks to be about 1/4 of a galaxy diameter away from Hosnian Prime (sorry, no image yet to prove that).
In-universe answer: technical mumbo-jumbo. MAYBE.
The novelization by Alan Dean Foster provides a possible explanation, from a certain point of view (the quoted paragraph was describing the operation of Starkiller weapon):
Traveling faster than anything ever generated by artificial means, through a torn portion of space-time whose properties were not fully understood, the concentrated glowing ball of energy lit the night sky above Republic City. Leia’s envoy Korr Sella was among those who gazed uncomprehendingly at the inexplicable phenomenon. Disturbed space was energized and lit up by its passage. It was as if a minuscule sun had suddenly appeared from nowhere, heading directly for the world on which she stood.
So, there's a possible somewhat "lasers make sounds in a vacuum" explanation, that somehow, that "not fully understood" space-time tear managed to allow people on Takodana to see the explosion of Hosnia Prime. Despite being (according to Visual Dictionary map) not only something like a quarter of a Galaxy away, but also not even on a direct path from Starkiller to Hosnian (sorry - I don't have a digital map yet, so you'll have to trust me at my word for now).
That, or Midichlorians made them see it.
Out of universe answer:
J.J. Abrams is hiding under a table ever since I asked him that question. I overheard him mumbling something about lens flares.
There's a chance that they'll retcon that in one of 17,214 new Disney
revenue sources canon books/comics/stories they planned. Always in motion, the future is.