Understanding growth function of closed intervals in $\mathbb{R}$


I as studying VCdimensions and growth functions and found the following example on Wikipedia:

The domain is the real like $ \mathbb{R}$ . The set H contains all the real intervals, i.e., all sets of form $ \{c \in [x_1, x_2] | x \in \mathbb{R}\}$ for some $ x_{0, 1} \in \mathbb{R}$ .

For any set C of m real numbers, the intersection $ H \cap C$ contains all runs of between 0 and m consecutive elements of C. The number of such runs of $ {m+1 \choose 2} + 1$ , so Growth(H, m) = $ {m+1 \choose 2} + 1$ .

Can anyone please explain to me what does the term "all runs of between 0 and m" refer to here and why the growth function is $ {m+1 \choose 2} + 1$ and not $ {m+1 \choose 2}$ ?

Thank you very much!