Nested summation - intuition

The essence is already encoded in the indices of the sums.

We can write for positive integer $r$ the sums as \begin{align*} \sum_{r_n=1}^r\sum_{r_{n-1}=1}^{r_n}\cdots\sum_{r_1=1}^{r_2}r_1\tag{1} &=\sum_{r_n=1}^r\sum_{r_{n-1}=1}^{r_n}\cdots\sum_{r_1=1}^{r_2}\sum_{r_0=1}^{r_1}1\\ &=\sum_{\color{blue}{1\leq r_0\leq r_1\leq \cdots\leq r_n\leq r}} 1\tag{2} \end{align*}

The number of summands given by the index range $$1\leq r_0\leq r_1\leq \cdots\leq r_n\leq r$$ is the number of ordered $(n+1)$-tupel $(r_0,\ldots,r_n)$ between $1$ and $r$ with repetition. This number is given by the binomial coefficient \begin{align*} \binom{(n+1)+r-1}{n+1}=\binom{n+r}{n+1} \end{align*} which corresponds to ($\star$) in OP's post.

These numbers are called Simplicial polytopic numbers.

$(1)$ corresponds to triangular numbers, $(2)$ to tetrahedral numbers, and so on. The relation with binomial coefficients and in fact to the whole Pascal triangle is well known.