Given two positive integers $c_1$ and $c_2$ such that $c_1 < c_2$, it's easy to prove that $n^{c_1}$ is $\mathcal{O}(n^{c_2})$ while $n^{c_2}$ is not $\mathcal{O}(n^{c_1})$. In general, for any functions $f(n)$ and $g(n)$ it is possible to determine which one (or both) is true, $f \in \mathcal{O}(g)$ or $g \in \mathcal{O}(f)$ using only definition of big-O notation. Thus it is not hard to see that TIME($n^{c_1}) \subsetneq$ TIME($n^{c_2})$ without using the Time Hierarchy theorem.
So, what is the meaning of the Time Hierarchy theorem and how and where can I use it?