Why can falling stars only be seen moving down towards horizon not up away from horizon? Are falling stars as we know only just space debris entering and burning up in the atmosphere and not a star at all?
The falling star or shooting star has nothing to do with a star. If shooting stars were actual stars, we'll be burning in their atmosphere, rather than the other way around.

When a meteoroid, comet or (small) asteroid (or space debris in some cases) enters the Earth's atmosphere at high speed (typically in excess of 20 km/s), aerodynamic heating produces a streak of light, both from the glowing object and the trail of glowing particles that it leaves in its wake. This is called a meteor or shooting star (or a falling star). 

In general, the meteors travel in a (approximately) straight path. We usually see only the large ones travel down towards the horizon while the small ones are burnt up long before that (giving the impression of coming from the horizon).



Image from Meteors? Even More Remarkable by Bruce Maccabee at brumac.8k.com

Consider the meteor path in the image above. it can be seen that,


The observer will see the meteor only after it passes the initial altitude $H_{i}$. If he sees the meteor at this point, and it burns up in the atmosphere, the impression is that the meteor comes from the horizon.
If the observer notices the meteor after it passes the initial height and tracks it through its travel in the atmosphere, till is exits at the final altitude $H_{f}$, it will appear as if the meteor is moving down towards the horizon.


Another factor is the thickness of the atmosphere itself. The thickness of the atmosphere increases in path of the meteor from $H_{i}$ to $H_{m}$, and decreases from there to $H_{f}$. As the 'glowing' of the shooting star is directly dependent on the density of the atmosphere, its visibility trails of near the observer's horizon. This is another reason for the meteor looking like its falling towards the horizon.

This can be seen clearly in case of meteor showers which appear to radiate from a point. 



Image from Lyrids Meteor Shower 2014 by Steve Owens at darkskydiary.wordpress.com

Here, the meteors appear to come away from the (radiant, which is above the) horizon.