When read in context this is self-evident (to me anyway)
Once you possess a creature’s body, you control it. Your game statistics are replaced by the statistics of the creature though you retain your alignment and your Intelligence, Wisdom, and Charisma scores. You retain the benefit of your own class features. If the target has any class levels, you can't use any of its class features.
You are in control of their body - the "Your game statistics ..." means the statistics of you in the body you are controlling (your body at the moment is a hunk of meat - their mind/soul is in the gem). Every statistic is that of their body except for what the spell says are yours so from the top:
- Hit points - theirs
- Armour Proficiencies are a class feature so - yours
- Ditto weapon proficiencies
- You get their hit points - your body (wherever it is) has your hit points and can be alive or dead
- If you are proficient in heavy armour in your own body then you are in theirs - if not then ... not
- Silver Swords - proficiency depends on your proficiency. As to getting the psychic damage and +3 bonus -
Silver Swords. In ancient times, gith knights created
special weapons to combat their mind flayer masters.
These silver swords channel the force of the wielder's
will, dealing psychic as well as physical damage. A
githyanki can't become a knight until it masters the
singular discipline needed to will such a blade into
existence. A silver sword is equivalent to a greatsword,
and takes on the properties of a +3 greatsword in the
hands of its creator.
It "... channel the force of the wielder's will ..." and "... takes on the properties of a +3 greatsword in the hands of its creator." The will involved here would be yours; does this allow you to deal the psychic damage? That would be a DM call - I am in two minds about it. The hands involved are physically the githyanki's but does it mean literal hands or metaphorical hands? That would be a DM call - I am in two minds about this too.
On thinking it over, I would rule that you do not get either benefit. My decision is based on considering the opposite situation of the githyanki using magic jar on your body - to my mind it can use its will in your body to activate all the properties of its sword. So if it can, you can't.