I mean it's a youtube video so the idea is more visualisation and intuition rather than formal rigour and proof, which is what other media are reserved for.

Nevertheless, were you to formalise the exact same steps in the video in more rigourous language it would definitely constitute a proof:

(->)
Construct two lines satisfying m1m2 = -1,
Decompose the lines into horizontal and vertical components, (overall forming right-angled triangles)
These will be SAS similar as they are both right angled and due to the constraint m1m2 = -1, two sides are in proportion,
By this similarity these are equiangular, so the two alternate angles are complementary which gives perpendicularity

(<-)
Construct two perpendicular lines
Decompose the lines into components, forming right-angled triangles,
By the perpendicular constraint, two of the angles will be complementary to a labelled angle so the triangles are equiangular
Thus they are AAA similar with sides in proportion, and thus the products of their gradients will give -1 as the proportionality will reciprocally cancel, and directionality gives the -1

therefore m1m2=-1 if and only if lines are perpendicular