Attached is a picture of a stiffener bead on a metal link, my question is how does this shape help stiffen the part? Why is this superior to simply having a flat metal link?
If I recall correctly, the added strength is due to the second moment of area (aka area moment of inertia).
Taking your example, assuming the end with the green dot is fixed and a load is applied to the other end, the dart will improve the link's resistance to deflection into and out of the screen.
Commenter jackwhirl is on the right path. And I will also say "if I recall correctly" because I haven't done this in decades. The equation for beam bending uses an area moment term for the cross section of the beam. Apply a ridge or other type of feature to the beam and beam stiffness increases. This is due to the area moment term of the cross section increases.
The further away from the bending axis the material the more contribution it makes to the stiffness. Material along the axis makes little contribution.
It's important to remember that this bracket will will not be loaded in the intended plane like in a FBD/theory. It will be have to have stiffness in its other degrees of freedom.
So by pressing some material out to the side you are adding torsional stiffness for a tiny loss of bending stiffness.
Edit: this looks suspiciously like a schoolwork type question.