How does sigmoid saturate with large weights?
The sigmoid function $$ \theta(z) = \frac{1}{1+e^{-z}}$$ looks like this :



where $$z=w_i a_i + bias$$ 
for activations $a_i$ from the previous layer, and weights $w_i$ of the current neuron.

When the weights $w_i$ are too large (positive or negative), $z$ tends to be large as well, driving the output of the sigmoid to the far left (value 0) or far right (value 1). These are saturation regions where the gradient/derivative is too small, slowing down learning.

Learning slows down when the gradient is small, because the weight upgrade of the network at each iteration is directly proportional to the gradient magnitude.