I don't remember the specific video but it was pretty elementary and got across the point that I had missed, you're not looking for a global optimum through some fancy operations on function spaces, you're just doing the old fashioned calculus thing of finding a maximum by setting a derivative to zero. Except you are doing that only at one endpoint of the mystery function, and its value (the boundary value) and derivative at that point (zero) are known, and you can work out the ODE that continues the solution. That's the Euler-Lagrange equation and suddenly everything makes sense.