Let S be a set of $n$ objects; then the binomial coefficient $(^n_k)$ is the number of $k$-elements subsets of $S$. Thus $\sum_{k=0}^n(^n_k)$ is the number of subsets of $S$ of all possible sizes from 0 through $n$.
The binomial theorem says that
\[(x+y)^n=\sum_{k=0}^n(^n_k)x^ky^{n-k}\quad;(1)\]
if you substitue $x=y=1$ in $(1)$ , you get
\[(1+1)^n=\sum_{k=0}^n(^n_k)1^k1^{n-k}=\sum_{k=0}^n(^n_k) \quad; (2)\]
And of course$(1+1)^n=2^n$, so $(2)$ reduces to
\[2^n=\sum_{k=0}^n(^n_k)={number\;of\;subsets\;of\;S}.\]
