God's anointing of Saul happened by Saul's meeting Samuel under God's arrangement and God's call through Samuel (vv. 11-27).
Jehovah had foretold to Samuel that He would send Saul to him, whom Samuel was to anoint as the leader over His people Israel (vv. 15-17). The day before Saul came, Jehovah had spoken to Samuel, saying, "At about this time tomorrow I will send to you a man from the land of Benjamin, and you shall anoint him as the leader over My people Israel" (v. 16a).
According to God's foretelling, Samuel told Saul that his father's donkeys had been found and that all the desire of Israel was for him and all his father's house (v. 20).
Saul was shocked by Samuel's word and answered, "Am I not a Benjamite, of the smallest of the tribes of Israel? And is not my family the least of all the families of the tribe of Benjamin? Why then do you speak to me in this way?" (v. 21; 15:17). Saul's humility here was a pretense. From the beginning he was a pretender. As we will see, when he knew that Samuel would appoint him to be king, he hid in "humility" (10:22). That he was not genuinely humble is seen in his being unwilling to give up the throne. Because he was altogether self-seeking, he tried desperately to keep the throne for his son Jonathan (20:30-31).
All the events recorded in 9:3-21 were under God's sovereignty. God caused the donkeys to be lost; He caused the servant to make a particular proposal; He revealed certain things to Samuel; and He sent Saul to Samuel. All these matters were used by God to accomplish something crucial—to make Saul king in order to discipline, to train, the children of Israel that they would learn not to have a king to replace God.