The Rakshasa stat block has this property:
Limited Magic Immunity. The rakshasa is immune to spells of 6th level
or lower unless it wishes to be affected. It has the advantage of saving
throws against all other spells and magical effects.
The spell Silence is written like this:
2nd-level illusion (ritual)
Since Silence is a 2nd level spell, so the Rakshasa is immune to its effects.
The Rakshasa is immune to the spell, therefore, he can still use spells with verbal components, speak and do an. However, the spell silence states:
Any creature or object entirely inside the sphere is immune to thunder damage,
The creature itself is affected not the Rakshasa, so the Rakshasa can cast a spell with thunder damage, but any creature within the sphere of silence is immune to the damage so will not take the damage.