Why do we create universal turing machines?
Theoretically, the universal Turing machine $U$ was defined to prove the undecidability of certain problems. Roughly speaking, we're talking about a machine that is able to decide what any other Turing machine decides. So assuming that Turing machines fully capture the notion of decidability, then it holds that anything that can be decided, can also be decided by $U$. Conversely, any problem that is not decided by the universal machine $U$ is considered to be undecidable.
To be more concrete, the ability of $U$ to take as input an encoding of another machine $M$ and an input $w$ of $M$, enables us (the machine $U$) to ask questions about other machines $M$. For example, a consequence of this ability is that we can ask whether $M$ halts on $w$: is there a universal machine $U$ that decides whether $M$ halts on $w$? More formally, is there a machine $U$ that decides the language $Halt_{TM} = \{ \langle M, w\rangle: M \text{ is a TM and $M$ halts on the input $w$}\}$.
One, with a diagonalization argument, can prove that the $Halt_{TM}$ is undecidable. $Halt_{TM}$ is known as the halting problem: is there a program that eventually halts and tells us whether another program (which is given as input) halts?
So, thanks to universal machines, we can ask all these interesting questions about machines, and get undecidability results.