It depends on where the point of silence was placed
Silence requires you to center the spell on a:
point you choose within range
How Resilient Sphere works is dependent upon where that point is.
If the point is inside the sphere, the wizard is silenced.
The point is within the resilient sphere and therefore the sphere wouldn't be able to stop the spell effects because it's already inside. In addition, the silence effects would be contained inside the sphere as well and the party would not be affected.
But the wizard isn't stuck in one place, so they are silenced until they move:
An enclosed creature can use its action to push against the sphere's walls and thus roll the sphere at up to half the creature's speed.
This would allow the wizard inside to move themselves away from the point of silence.
Do note that other creatures can move the globe as well, and may be able to move the point of origin back inside the sphere:
Similarly, the globe can be picked up and moved by other creatures.
If the point is outside, the wizard is no longer silenced.
Resilient Sphere states:
Nothing--not physical objects, energy, or other spell effects--can pass through the barrier, in or out, though a creature in the sphere can breathe there.
If the spell effect's point of origin is outside (or the wizard is outside the range of it), the sphere actively prevents spell effects from passing through. The inside of the sphere would be a safe bubble from silence.