Learn Turkish: Sizden bir şey rica ediyorum. - I am requesting something from you.

The suffix -den means "from". So sizden means "from you" (formal or plural). In English, we might ask something "of" someone, but in Turkish, you always request a thing "from" a person.

Is bir şey always written as two separate words?

Yes! The word şey means "thing", and a strict spelling rule in Turkish is that şey is always written separately from the word before it. So "something" is always bir şey, never "birşey".

The base verb is etmek, so why does it change to ediyorum here?

When the present continuous suffix -iyor is added to etmek, the letter t softens to a d because it is sandwiched between two vowels (e and i). This consonant mutation makes the word easier to pronounce.

I've heard rica ederim used to mean "you're welcome". How does that relate to this sentence?

They use the same verb! Literally, rica etmek means "to make a request". When someone thanks you, replying with the simple present form rica ederim essentially means "I request (that you don't mention it)". In this sentence, the continuous form rica ediyorum is used for its literal meaning: "I am making a request".