Sub-hyperspace
The Hosnian Cataclysm was observable in the skies of far-off planets, including Takodana, because of a tear in what is known as sub-hyperspace.
Sub-hyperspace is what the First Order called a rupture in the space-time continuum that served as a conduit for phantom energy. Unlike standard hyperspace, which traverses around the galaxy, sub-hyperspace cuts through it. Starkiller Base, under the control of the First Order, had the capability to harness a type of dark energy referred to as quintessence. It could then transform this quintessence into phantom energy and project it along a straight path via sub-hyperspace. This process allowed them to obliterate entire star systems across immense interstellar distances instantaneously.
Behind the scenes
Pablo Hidalgo stated in a tweet that the massive amounts of energy unleashed when the Starkiller weapon was fired could create a temporary breach in sub-hyperspace. This is what made it possible to witness the destruction of the Hosnian system from across the galaxy as it unfolded.