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.
A geometric algebra is a vector space where the elements are products of vectors. Suppose, for example, we are working with a 3D Euclidian orthonormal basis $ \left\{ {\mathbf{e}\_1, \mathbf{e}\_2, \mathbf{e}\_3 } \right\} $. Then the following would all be elements of the geometric algebra that is said to be generated by this basis
$$\begin{aligned} &\mathbf{e}\_1 \mathbf{e}\_1 \\ &\mathbf{e}\_1 \\ &\mathbf{e}\_1 \mathbf{e}\_2 \\ &\mathbf{e}\_1 \mathbf{e}\_1 \mathbf{e}\_1 \mathbf{e}\_2 \mathbf{e}\_3.\end{aligned}$$

In addition to the usual vector space axioms, a geometric algebra has a contraction axiom, defining the product of any vector with itself

$$\mathbf{x}^2 = \mathbf{x} \cdot \mathbf{x}.$$

The multivector formed by adding all of the example products above, after reduction using the contraction axiom, is
$$ 1 + \mathbf{e}\_1 + \mathbf{e}\_1 \mathbf{e}\_2 + \mathbf{e}\_1 \mathbf{e}\_2 \mathbf{e}\_3,$$ which happens to have an element of each grade, a scalar, vector, bivector, and trivector.