Where do you see cyclic quadrilaterals in real life? I've just been studying cyclic quads in geometry at school and I'm thinking see seems pretty interesting, but where would I actually find these in the real world? They seem pretty useless to me...
 A: I can't think of any applications, and I doubt any satisfactory ones exist - for example, as noted in the comments there may well have been connections to astronomy, but I think it's fair to suggest that almost no-one who is being taught circle theorems is going to use them in their life at any point.  
Thus, I'm going to interpret this question as:

Why would you bother learning a theorem that has no application in real life?  

And I think there are two good answers to this question:
1. It is interesting
This is, really, the only reason you're taught anything in your life other than how to pay taxes. Geometry is something that lots of people over a long period of time have found to be intrinsically interesting. The reasons for this are complicated - it's a good intellectual exercise, and for many people intellectual exercises are something they enjoy doing.  
2. It forces you to think logically
The patterns of thought people generally use in mathematics are valuable. Logical arguments are important in all walks of life, and being able to understand and interpret them is an extremely valuable life skill which you really should want to have.  
I have a lot of sympathy with this question, for the following reason: you are probably taught mathematics very badly. The arguments I give above really rely on the idea that you are taught how to prove theorems (and Euclidean geometry is a fantastic exercise in proof). Without that, I would claim that learning geometry really has no value. I would even go so far as to say you shouldn't bother going so far as to learn basic trigonometry (unless you need it to be an engineer or something), unless you study its proof. That really is where all the value, and all the fun, is.  
This is not your fault. But there is something you can do about it. Look up a proof, try to understand it, and if you're lucky you'll get a little intellectual buzz from the 'aha!' moment of it all coming together. But, I'm sorry to say, you'll probably have to do this yourself. Mathematics teaching is woeful in the vast majority of schools, and statistically speaking you are unlikely to even have a teacher capable of explaining to you why these results are true, let alone interesting or useful.  
So, on the off chance that this answer has spiked your curiosity, I recommend writing another question, called "How do you prove interesting facts about cyclic quadrilaterals?", and you might get a more satisfying answer.
A: Theorem 3 of the Bern-Eppstein paper cited below proves that any polygon of $n$ vertices
may be partitioned into $O(n)$ cyclic quadrilaterals.
A hint of how this might be achieved can be glimpsed in the figure below, where
all the white quadrilaterals are cyclic.

          


          

Fig.5 from cited paper.


Quadrilateral meshing is important in many applications. The cyclic
quads produced by their algorithm have desirable "quality" characteristics.


Bern, Marshall, and David Eppstein. "Quadrilateral meshing by circle packing." International Journal of Computational Geometry & Applications 10.04 (2000): 347-360. (Pre-journal arXiv abstract.)
  (Journal link.)

A: Ptolemy's theorem says that if $a,b,c,d$ are the lengths of the sides of a cyclic quadrilateral, with $a$ opposite $c$ and $b$ opposite $d$, and $e,f$ are the diagonals, then $ac+ bd = ef$.  In the second century AD, Ptolemy used that to prove identities that today we would express as
\begin{align}
\sin(a+b) & = \sin a \cos b + \cos a \sin b, \\
\cos(a+b) & = \cos a \cos b - \sin a \sin b.
\end{align}
As to where these come up in Reality, you can start with this: https://en.wikipedia.org/wiki/Uses_of_trigonometry
PS: A bit more on what Ptolemy did: https://en.wikipedia.org/wiki/Ptolemy%27s_table_of_chords
