What does it mean when they say Trigonometric Ratios of "Standard Angle"? First, let me be clear. I understand what is a "Standard Angle" or "Angle in Standard Position", my question is why are we interested in the Trig Ratios of only standard angles? How does non-standard angle affect the result? 
PS : I'm a high schooler, starting out with Trig. Forgive me if this question sounds dumb. I just can't seem to accept why only "Standard Angles"?
 A: You ask interesting questions, but if you have a good teacher I believe he should be able to explain such things (BTW, have you considered studying mathematics beyond the secondary level? You should love it).

Alright, for what you have called a standard angle, you should understand that when trying to use numbers in geometry, we want to ideally have a name for every point we care about -- and that is usually every point in some space -- e.g., a line, plane, etc.
Let us start with a line. To name the points on a line, we could simply use any open interval of real numbers of our choice, and pick any number as our origin. However, as I said in response to one other related question of yours, why overcomplicate matters when it doesn't really matter? -- so we simply allow our interval to contain $0,$ which we take as our origin. Now, if we want a nice symmetry, which always simplifies stuff (mathematicians, like other people, prefer to take the path of least action, when possible), we may allow the interval to be of the form $(-r,r).$ We may now import our knowledge of the behaviour of numbers in this interval to talk about linear points (usually, we allow ourselves to include all possible real numbers). Since a plane can be thought of as the product of two lines, it is easy to see that we can use ordered pairs of such lines as we've defined to name every single planar point (we can continue this scheme in the obvious way to higher dimensional euclidean (flat or straight) spaces).
If we work in such a co-ordinated plane (and we do because it leads to discoveries we couldn't have so easily made otherwise), and want to continue talking about angles (and we do, for reasons I hope you may now at least sufficiently appreciate), then it is clear that we can move any angle we wish to talk about so that its vertex coincides with the origin (why should we want this -- well, because we can, and why overcomplicate matters for ourselves when we do have a choice?). All this is clear.
Now why on earth do we need to fix one ray of the angle? Well, for similar reasons that we fixed a unique point as origin, namely that we want some symmetry, which is always nice if we can help it. Again, to talk about an angle (by which I now mean a rotation), you can see again that we need infinitely many names since you can imagine an object rotating about a point in the plane for ever and ever, whether clockwise or counterclockwise (indeed you may think of two rays from a common point, one rotating clockwise and the other counter forevermore). Again, since we want to talk about every possible rotation, it is clear that the real numbers will come in handy. However, we would have to fix a ray as the original position of our rotating ray. We could call this original rotation any number, but again we usually choose to name it $0$ because  it simplifies things for us.
With a ray emanating from the origin, we could describe all possible turns using the reals, taking one direction as positive and the other as negative.
