Why is it "yours faithfully" and not "your faithfully"?

If the valediction ends with a noun, then the s is omitted, as in:

Your obedient and humble servant,
Your friend,

If the phrase uses an adverb, then yours is used:

Yours truly,
Sincerely yours,
Yours forever,

There's nothing tricky about this; just think about how you'd say the full sentence, if it began with
“I am...”

I am your closest ally.
I am truly yours.

In the 18th and 19th centuries, many letters ended with the full sentence. More here.

It's an inversion of I am faithfully yours

This is your cup = it is yours

I am your faithful [servant](various traditional terms could apply) = I am faithfully yours

It is just standard usage. Alternatives are "Yours sincerely" or "Sincerely yours" or "Yours truly".

Without the s something like:

"Your faithful admirer"