Can anyone help explain what bijection is in the context of this problem, and how exactly it's used to derive the particular solution
From the definition provided it seems as though a bijection is a translation provided to a set of points, but I'm not sure that I understand how that is applied, especially in the context of the problem. Thanks!
Let's look at a smaller example. Suppose there are $7$ spaces in the parking lot, and $4$ cars arrive.
On the left, we list all of the ways that four cars can park so that there are no two adjacent empty spaces. On the right, we list the results of following the instructions in the solution: between each pair of consecutive empty spaces, delete one of the intervening cars.
On the left, we have a a complicated object, which is unclear how to count. On the right, we see something simpler; every possible way to park $2$ cars in a row of $5$ spots is listed exactly once. The number of such arrangements on the right is clearly $\binom{5}2$. Since the left and right columns have the same number of arrangements, this also counts the number of arrangements on the left.
This is the purpose of clever bijections. You make a perfect matching between a mysterious set and a simple one, and use what you know about the simple one to explain the mysterious one.