Notice that the sum of the powers of $2$ from $1$, which is $2^0$, to $2^{n-1}$ is equal to ${2^n}{-1}$. Please explain in quotations!
"Notice that the sum of the powers of $2$ from $1$, which is $2^0$, to $2^{n-1}$ is equal to $2^n-1$." In a very simple case, for $n = 3, 1 + 2 + 4 = 7 = 8 - 1$.

Since we don't know what $1+2+4+\cdots+2^{n-1}$ is yet, we'll call that number $S$. So we have $S=1+\cdots+2^{n-1}$. Multiplying by $2$ on both sides, we get
$$
2S=2+4+8+\cdots+2^{n-1}+2^n
$$
Here comes the trick: calculate $2S-S$:
$$
\begin{align}
2S-S=&2+4+8+\cdots+2^{n-1}+2^n\\-(1+&2+4+8+\cdots+2^{n-1})
\end{align}
$$
and we see that all but two of the terms disappear, and we're left with $2S-S=2^n-1$.
