I was taking Stanford's cs231n class and was unable to understand the gradient calculated using the SVM loss function.

You should go here to check the notes which I am talking about.

This is the SVM loss function. And now we want to find it's gradients. With respect to the weights of the correct class, the gradient of the loss function will be, Until Now I understand Everything, the problem occurs in finding the gradient with respect to the incorrect classes : I do not understand the calculation of this gradient, where did the Sigma go?

When you take the derivative wrt some $$k\neq y_i$$, the $$w_k$$ appears in the whole expression just once, i.e. when $$j=k$$. And, the gradient will be just $$x_i$$ times the indicator.
For example, let the set $$j\neq y_i$$ be $$\{a,b,k\}$$. The expanded version of the loss function will be $$L_i=\max(0,w_ax_i-w_{y_i}x_i+\Delta)+\max(0,w_bx_i-w_{y_i}x_i+\Delta)+\max(0,w_kx_i-w_{y_i}x_i+\Delta)$$
If you take the derivative with respect to $$w_k$$, you'll have just one indicator function, leaving you with no sigma expression.