Number of subsets of length 7 I have the following summation:
$$\sum\limits_{k=7}^{n}  {k-1\choose 6} $$
and apparently it counts the number of subsets of {1, 2, . . . , n} having size 7.
Why is this? 

It is because of Pascal's recurrence.  The recurrence is that $\binom{i}{j}+\binom{i}{j-1} = \binom{i+1}{j}$ provided that $0< j\leq i$.
\begin{align*}
\binom{6}{6}+\binom{7}{6} &= \binom{7}{7} + \binom{7}{6} \\
                          &= \binom{8}{7} \\
\binom{6}{6}+\binom{7}{6} + \binom{8}{6} &= \binom{8}{7} + \binom{8}{6} \\
                          &= \binom{9}{7} \\
\binom{6}{6} + \binom{7}{6} + \binom{8}{6} + \binom{9}{6} &= \binom{9}{7} + \binom{9}{6} \\
&= \binom{10}{7}
\end{align*}
This process repeats until you get to $\binom{n-1}{6}+\binom{n-1}{7}=\binom{n}{7}$.  It helps to draw Pascal's Triangle and follow the additions down the tree...
