Why the region has to be convex

The simplex algorithm only works when the feasible region has a particular shape. Pick any two points on the outside of the region. The straight line joining them must stay entirely inside the region without being interrupted. A shape with this property is called convex.

This comes directly from the equations being linear. Linear constraints produce flat edges and faces, which is what keeps the region convex. It also means the variables cannot appear as exponents. If a constraint used $c^2$ or $d^3$, the boundary would curve and the method would break.

A dented or crescent shape is the failure case. There you can find two boundary points whose connecting line pokes outside the shape, and the simplex algorithm, which reasons by moving along edges between vertices, can get the wrong answer. So the requirements stack up: linear equations, a convex feasible region, and non-negative variables. Keep those and the vertex-walking logic from the main-ideas lecture is guaranteed to make sense.