Those who consider themselves religious and yet do not keep a tight rein on their tongues deceive themselves, and their religion is worthless.
– James (Jesus' brother) 1:26
Why? What happens when we're silent other than the annoying chaos of our thoughts? The answers to these questions are always unfolding and completely unique to each individual's practice and can only truly be understood intimately by each person, I think. With that having been said, there are common generalities.
Silence opens us to savoring. Too often, we come into revelation and instantly start speaking about it. This practice leaves the depth of our understanding on the surface level of mere encounter and excitement.
I imagine an illustration for this being something like my meeting Shawna for the first time. Very exciting! She is captivating! And very beautiful! Then, it would be like if I turned around and spent the rest of my life describing her to everyone I met. The depth of my relationship with her would only be in my description of her. I wouldn't know her. And she wouldn't know me (and, truth be told, what she might know of me I doubt she'd like very much).
Setting aside time to be silent exposes us to a greater depth in what others are saying, what our circumstances are saying, what our own hearts are sub-consciously saying, and (usually, after we've sorted through all those other voices) what God is saying in the midst of it all.
When we take our time to be silent, we take our time to speak. If Jesus' brother James has any clout with you, it could be worth exploring how silence will effect your faith.