removed him from the kingship

This expression means God caused Saul to stop being king. Alternate translation: "rejected Saul from being king"

he raised up David to be their king

"God chose David to be their king"

"the king of Israel" or "the king over the Israelites"

"God said this about David"

"I have observed that"

This expression means he "is a man who wants what I want." (See: rc://en/ta/man/translate/figs-idiom)