Question: I went through some introductory videos of this topic and this was an idea presented in each. Exactly, what does geometric algebra do different than linear algebra which makes this operation sensible?

Point is mentioned at 11:52 of this [video](https://youtu.be/60z_hpEAtD8?t=709) right after introducing the geometric product of how bivectors and scalars can be added, and, again at 17:58 where a multivector with scalar, vector and bivector components is shown

Answer: 
This is from my book:

How can we add, e.g, a scalar and a vector? Are we not adding apples and oranges? Yes, but there is a sense in which we *can* add apples and oranges: put them together in a bag. The apples and oranges retain their separate identities, but there are “apples + oranges” in the bag.
The short answer is something like "because you can define anything you want in mathematics" (the [video](https://www.youtube.com/watch?v=60z_hpEAtD8&t=709s) linked in the [OP](https://www.urbandictionary.com/define.php?term=OP#5393886)). And the long answer(s) depend on a choice of foundations of math that 1. was not specified in the question and 2. the vast majority of mathematicians don't have to worry about. I'll try to [thread the needle](https://en.wiktionary.org/wiki/thread_the_needle#Verb), though.

Suppose, for example, that we're doing geometric algebra in two-dimensional space. Then every vector is of the form $b\mathbf i+c\mathbf j$ for scalars $b,c$; and every bivector is of the form $d(\mathbf i\wedge \mathbf j)$ for some scalar $d$. So we can consider lists of four scalars as the objects in $2\text{d}$ geometric algebra, with $\left\langle a,b,c,d\right\rangle$ being written as $a+b\mathbf i+c\mathbf j+d(\mathbf i\wedge \mathbf j)$, and any complicated expression with scalars, vectors, and bivectors able to be expanded out into the individual $\mathbf i$ and $\mathbf j$ components, so that we could identify the four scalars in the "list of four scalars" representation, if we wanted to.

In general, we can handle $n\text{d}$ geometric algebra with lists of $\binom{n}{0}+\binom{n}{1}+\cdots+\binom{n}{n-1}+\binom{n}{n}=2^n$ scalars.