In general, you solve a homogeneous ODE with constant coefficients by solving the "characteristic equation". For a second order equation, with real coefficients, that will be a quadratic equation with roots of the form a+ bi and a- bi. The corresponding solutions to the ODE will be [itex]e^{(a+ bi)x}= e^axe^{bix}[/itex] and [itex]e^{(a- bi)x}= e^xe^{bix}[/itex]. As you have probably learned, [itex]e^{bix}= cos(bx)+ i sin(bx)[/itex] (you can get that by comparing the Tayor's series of [itex]e^{bix}[/itex] with the Taylor's series of sin(bx) and cos(bx)). In particular, sin(2x) and cos(2x) correspond to [itex]e^{2ix}[/itex] and [itex]e^{-2ix}[/itex] which means that the "characteristic solutions" are 2i and -2i which, in turn, come from the characterstic equation [itex]r^2= -4[/itex] or [itex]r^2+ 4= 0[/itex] which corresponds to the ODE [itex]y''+ 4y= 0[/itex]. The point is that, to use the "Annihilator method" (also called the "method of undetermined coefficients"), the function of x, here, sin(2x), must be a possible solution of a homogeneous differential equation with constant coefficients and that is true here- sin(2x) is a solution to y''+ 4y= 0.
As for "you skipped finding and applying the actual annihilator", I watched this entire video and did not find those words.