Why is it that $5$ vectors of $\Bbb R^6$ can't span $\Bbb R^6$ but $6$ vectors of $\Bbb R^5$ can span $\Bbb R^5$?
The first case would mean a $6\times 5$ matrix and the second case would be a $5\times 6$ matrix so the latter has more columns than rows so it has to be linearly dependent hence it cannot span $\Bbb R^5$?
You are mixing up concepts. Usually, we use the term vector to denote an element of a vector space, such as $\mathbb{R}^5$, while a matrix is a linear map between two vector spaces (at least, its representation in a certain basis). They are radically different things, but they are often introduced in the same framework and so people get confused.
What does it mean that certain vectors span a vector space? Well, it simply means that you can start from the origin of your vector space and get to any other point just using those vectors (where you are allowed to multiply them by scalars and sum them together).
Now, say you are in $\mathbb{R}^6$. Then intuitively you have $6$ directions in which you can move. If you are given only $5$ vectors, there is no hope of getting to every point of the space just using them, as they give you at most $5$ possible directions. On the contrary, if you have $6$ vectors in $\mathbb{R}^5$, they can very well give you all $5$ possible directions (and you will moreover have some repetition, which correspond to the fact that the vectors will not be linearly independent). Thus, $6$ vectors in $\mathbb{R}^5$ can span the whole space.
so the latter has more columns than rows so it has to be linearly dependent hence it cannot span $\Bbb R^5$?
You are right that 6 vectors in $\mathbb{R^5}$ will definitely be linearly dependent, but that doesn't mean they cannot span $\mathbb{R^5}$. It means they can't form a basis because to be a basis, you need the vectors to not only span $\mathbb{R^5}$, but also be linearly independent. The question here however is not if they form a basis or not, but if they can span $\mathbb{R^5}$.
Since $\mathbb{R^5}$ has dimension 5, you need at least 5 vectors to span the space. Note that having 5 or more vectors does not guarantee they span $\mathbb{R^5}$, but the question here is only if they can.
On the other hand, $\mathbb{R^6}$ has dimension 6 so you need at least 6 vectors to span the space; therefore 5 vectors definitely won't span it.
Saying ‘$6$ vectors of $\mathbf R^5$ span $\mathbf R^5$’ does not mean they do for any set of $5$ vectors.
For they first assertion, it is true by the following characterisation of the dimension of a (finite dimensional) vector space: it is