Don't rush a person to share
One of the fundamental tools in counselling is time and space. Space for the client to settle their feelings sufficiently to find words to describe them. If you rush somebody already in emotional distress what they start to express is something more than or different to what it was bothering them. They may now become irritated or stressed when they were originally upset. And so this becomes even more confusing and overwhelming.
It is usually the listener that feels discomfort with the silence. They may feel they should be doing something or have their own general unease with silence and stillness.
Doing something can be saying something like, "I am here for you when you are ready". This does not mean you go off and do something else while you wait. Sit patiently and attentively with them. If you do something else while you wait, like browse your phone, they may interpret this as them being a burden to your time and thus puts even more pressure on them. However, it may take the pressure off the silence if you are only doing something casually whilst still ensuring they know you are there. You could ask them if they want you to do something for a while and come back to them. If they don't know chances are they want you to stay but feel guilty I doing so. So reassure them you are ok just sitting with them in the silence.