Hi hsumail, thanks for the question, and apologies for taking so long to get back. Conservation of energy is being used here, though it could be expressed as the work-energy theorem as well and get the same result. The idea is that the total energy before a process is the same as the total energy after the process, and the formulae surrounding this subject serve just to help account for all the different forms of energy before and after the processes. Conservation of energy problems are book-keeping exercises.

Now I'll get to your comment. You mentioned the work-energy theorem, so let's clarify that it's $W_{ext} = \Delta KE$, not the expression written in your comment. [edit: earlier comments about the Earth don't apply. I hadn't actually looked up the problem.] You could consider the car-spring as a single system, which is all well and good, but this isn't strategic for problem solving since your only conclusion would be that $W_{ext} = 0$. This problem is asking for the spring constant, and conservation of energy is the ticket to solving for it. Maybe you would prefer the following expression, rather than the one I wrote in the video:
$KE_1 + PE_1 + W_{NC} = KE_2 + PE_2$. $W_{NC}$ typically represents friction, which is zero in this case, so we plug different types of energy into this expression and get $\dfrac{1}{2}mv_i^2 + 0 + 0 = 0 + \dfrac{1}{2}kx^2$ where I've substituted specific expressions in the same order as the equation before that has more generic expressions. This will arrive at $\dfrac{1}{2}mv_i^2 = \dfrac{1}{2}kx^2$, which is the same as in the video.

Hope this helps,
Mr. Dychko