"Samuel took the vial of oil and poured it upon his head, and he kissed him and said, Has not Jehovah anointed you as prince over His inheritance?" (10:1). Here we see that Saul was made king by Samuel's anointing him with oil.
In 10:2-13 we are told how God's anointing of Saul to be the king was confirmed.
Samuel told Saul that he would encounter a group of prophets who would be prophesying and that the Spirit of Jehovah would rush upon him (vv. 5b-6a). This word was fulfilled later that day when a group of prophets met him. Then the Spirit of God rushed upon him, and he began to prophesy among them (vv. 10-13).
Samuel also told Saul that after the Spirit of God rushed upon him, he would "be turned into another man" (v. 6b). This word was later fulfilled (v. 9a).
When Saul's uncle checked with Saul where he had gone with his servant, Saul told him that they had gone to look for the donkeys, and when they were not to be found, they went to Samuel (v. 14). Then Saul's uncle said that he should tell him what Samuel said to him. Saul told him the part about the donkeys, but "he did not tell him about the matter of the kingdom that Samuel had spoken of" (vv. 15-16).