How did we derive this general term for the series? We have this series of numbers:
$1, 3, 6, 10, 15$
The general term can be described wit: $\frac{r(r + 1)}{2}$
Apparently the following series:
$1, 4, 10, 20, 35$
Can be described with $\frac{r(r + 1)(r + 2)}{6}$ based on the first series.
But I am not clear how this is derived. Can someone please explain?

Those aren't series, they are starts of sequences.
As always when given the start of a sequence, it can continue in any way so there's infinitely many formulas that will describe give a sequence that starts with the given numbers. We can only try to find a simple formula and hope that was what was though of.
Those sequences are well-known, so a lot of people will be able to just give the formulas you have.
If we assume that they might be generated from polynomials of a degree, we can calculate the differences between consecutive elements. For the first sequence this gives:
$$
2,3,4,5
$$
and if we iterate:
$$
1,1,1
$$
So after two steps we got a constant sequence, so it's a second degree polynomial $an^2+bn+c$.
If you do the same with the second sequence, you'll find it takes three steps until you get a constant sequence so the values comes from a third degree polynomial $an^3+bn^2+cn+d$.
The coefficients can be derived from the differences, or you can plug in a few of the known values and get a set of equations for the coefficients.
