This is explained in the film's Junior Novelisation .
The super-weapon is capable of firing a beam that travels faster than light.
Evidently it also creates an effect that can be seen faster than light, explaining why people in other systems can see it in realtime: Outside the castle with Chewbacca, Han surveyed a section of Takodana’s sky through a compact ponipin telescope.
He’d heard many theories about the origin of the star that had recently blinked into existence.
None made any sense.
New stellar bodies didn’t just brighten the sky all of a sudden.
Han’s ponipin measured that the new star was also many, many light-years away, which under normal astronomical circumstances meant that it had actually blazed to life years before.
Moreover, if the calculations proved correct, the stellar coordinates happened to be the same as those of the Hosnian system, where the capital of the New Republic was located.
Could the Hosnian system have gone nova?
Could it—dare he even consider—have been destroyed?
And if so, how had it happened so quickly?
This was confirmed in a tweet by the film's Creative Executive, Pablo Hidalgo
Apparently the explosions were... ...
Some weird-ass hyperspace-rip seen-across-the-cosmos kinda shit.
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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.
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From all I can see, it's chalked up to artistic license.
The systems are not so close that they would see the attack; it was shown purely for dramatic effect.
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This is likely among the biggest logistical problems in regard to space physics in the Star Wars film universe thus far (up there with the visibly-spinning "galaxy" at the end of Episode V).
I didn't have the benefit of the novelization's explanation at the time of my viewing; as a moviegoer, here's how I reconciled the events at the time I watched the film from a scientific perspective:
All orbs in the "Hosnian System" were so close together visibly that it strained credulity (at least for me) to think that they were all individual planets, but instead a "system" of celestial bodies that included a main planet and natural satellites within a larger star system (like how we call Jupiter and its moons the "Jovian System").
It would seem to me that the best retcon to fix this plot hole would be to have "Hosnian Prime" and its satellites lie within the same star system as Takodana.
That way, the propagation delay between its destruction and the visibility thereof from Takodana could be a matter of minutes rather than millennia .
That would be an elegant retcon that would only require the filmmakers to rewrite a map, but it appears as though the novelization's explanation (mentioned above) is the most official answer offered thus far.
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It's clear that spacetime just works differently in Star Wars.
The millennium falcon can jump from system to system just by jumping to light speed.
So either light moves faster, time moves faster (as seen from our perspective, it explains the visibly rotating galaxy somewhat) or the Galaxy far far away is just very small.
Same can be said of other things.
Putting the mass of a sun in a rocky planet?
It's a fantasy universe.
There is even magic in it (the force).
So trying to put it into a scientific perfection is not relevant.
This is the reason the midi chlorians sucked.
The force is not scientific.
Nothing is Star Wars is!
Furthermore the New Order wanted people to see and fear the destruction.
So the projection of that destruction in other systems could be intentional.
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On Takodana,    the main characters and various bar patrons can see the planet Hosnian Prime being destroyed by Starkiller Base with their (respective) naked eyes.  How did they see this? Is the Hosnian system close enough to see?  If it's not very close and they could just see the explosion, how did the light from it reach them so quickly?
How is Starkiller Base's beam visible from Takodana in The Force Awakens?
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<star-wars><the-force-awakens>
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It is generally accepted that the Hosnian system is not very close to Takodana. However, the superweapon used by the Starkiller Base is capable of firing a beam that travels faster than light. Therefore, any effects of this beam would also be seen faster than light, explaining why people in other systems would be able to see the destruction in real time. There a number of possible explanations - the most authoritative comes from the film's Creative Director Pablo Hidalgo who explained that it is because the super-weapon fires a beam that is faster than ligjht.
The film's Creative Director Pablo Hidalgo has confirmed that this was because of a super-weapon firing a beam that travels faster than light. It is not at all close but it may appear to be due to complex technical or physical processes at play, as suggested in the novelisation by Alan Dean Foster. It's just artistic license - done for dramatic effect. One possible explanation is that the orbs in the Hosnian System are actually a planet and its satellites rather than a star and its planets. It is because spacetime works differently in the Star Wars universe.