Why does being bounded below in Swish reduces overfitting In the Swish activation function paper here, it reads:

Being bounded below may be advantageous because of strong regularization — functions that approach zero in a limit to negative infinity are great at regularization because large negative inputs are discarded. This is important at the beginning of training when large negative activation inputs are common.

I think that the common ReLU is also bounded below, but why does being bounded below reduces overfitting (strong regularization)? I cannot understand the idea behind.

Indeed relu is also bounded below, they didn't claim otherwise. The difference is, that swish allows small negative values for small negative inputs, which according to them, increases expressivity and improve gradient flow.
The reason behind improving generalization is that, as in regularization, small, approaching zero, weights improve generalization as the function become more smooth and it reduces the effect of fitting the noise.
They claim that by bounding large negative vales in the activation function, the effect is that the network "forgets" large negative inputs and thus helping the weights to approach to zero.
See the image they added, large negative values, which are common before training are forgotten and after training the negative scale is much smaller.

There is a tradeoff between bounded which improve generaliztion and unbounded that avoids saturation of gradients, and help the network to stay in the linear regime.
