First off, let me just mention that probably the easiest way to prove the identity:$$a^3+b^3+c^3 - 3abc = (a+b+c)(a^2+b^2+c^2 - ab - ac - bc)$$is to just expand everything on the right-hand side and find that it matches up.

That being said, it can be good to practice with some other identities and prove the formula as done in the video lecture. When you get to $(a+b)^3-3ab(a+b+c)+c^3$ note that $(a+b)^3 + c^3$ is the sum of two cubes. So this portion can be factored using the sum of cubes formula, which turns it into  $(a+b+c)*[(a+b)^2-c(a+b)+c^2]$. Hope this helps!