Samuel wants to say that there are two threads that run through the events of recent times. The first thread is that of the people, who, through distrust choose and covet against the kingship of God. This is in his mind when he says: here is the king you have chosen, the one you asked for. But the other thread is that of God. He had his own exalted goal with that kingship, the kingship fitted into his plan for Israel. Noting this, Samuel says: see, the Lord has set a king over you. Because of the latter the king cannot go away, the matter cannot be turned back. The king remains. But that does not mean on the other hand that it is to be forgotten, spoken of no more. On the contrary, Samuel sharply points out the people’s thanklessness and brings it into the light.
That is the goal of the historical exposé, where Samuel makes the nation look back to the past, to the righteous deeds that God has done for them and for their fathers (1 Samuel 12:7). The righteous deeds, that is to say: the Lord God kept the promises he made in his holy covenant. Just like everyone must acknowledge, when they look back, that their life is full of these righteous deeds of the Lord; that is to say, that he was present and made his presence known. Each time there was a crisis, even though this may have been through their own doing, the Lord saved, he delivered out of the hand of the enemy. Read about this in the book of Judges: Gideon, Barak, Jephthah...
That is what Samuel is aiming for: that Israel shall be ashamed of its ungratefulness and, together with their king, shall walk in the fear of the Lord.
13 And now behold the king whom you have chosen, for whom you have asked; behold, the LORD has set a king over you.