There are two uses silence has in Artifact. Firstly, it prevents abilities from being used. This goes for hero cards and creeps, but also for equipped items that have an ability. The second effect is less obvious from the word ‘silence’ itself, but shouldn’t be underestimated either. A silenced hero card ‘loses’ its color for the duration of the silence. This means that the hero will no longer function as a gateway for spells, creeps and improvements, who all require a hero of their same color to be in a lane in order to be played.
Only a couple of cards that can silence has been revealed so far. Rix’ premier card Truth to Power is perhaps as basic as it gets when it comes to card texts, and silences a single unit for a round. Then, we have Bloodseeker's Blood Rage which silences a unit in exchange for increased attack.
Silence can be a real nuisance for your opponent if you use it well. If you fear an ability will be triggered and will give you a significant disadvantage, silencing the unit will keep you save for at least another turn. But that’s nothing compared to being able to prevent your opponent from playing an entire color. If your opponent has a single hero of a specific color in a lane and you expect a powerful creep, spell or improvement to be in your opponent’s hand, you can effectively lock that card.
If you suspect your opponent to have a silence effect there are two paths to take. You can change the order of the actions you wanted to take and trigger abilities first, before they can be silenced. The other thing you can do — which applies more to silence’s secondary effect — is wait the game out a bit and give your opponent the impression that you don’t have a specific card in hand. This little mind game might trick your opponent into not silencing a certain hero, which then gives you the opportunity to play your power card.