I am not familiar with what modern Lutherans might believe, if any different from Luther, but I am very familiar with what Calvin and Luther thought. They thought the same in many ways but used different lenses. I do not see them at disagreement. They differed in emphasis and detail. Luther emphasized Christ dying for all (which Calvin also believed). Calvin emphasized election more than Luther (which Luther believed).
What makes them seem different is that Calvin speculated deeper on the subject where Luther does not really form an opinion. Calvin sees the death of sin as like physical debts. Son can’t be aid for twice, therefore sins are either paid for by Christ or by the sinner. In Calvin’s view Christ only died effectually for sin. In Luther’s Christ died provisionally for all sin and effectually only for those that receive it. In other words Christ paid the penalty and then the sinner pays, so the sin is paid for twice – under a Calvinist view of Luther.
Luther himself would not draw this Calvinistic distinction and was happier leaving something hidden behind a veil of the unknowable. The end result is the same.
As far as the recording you had linked to I do not find the Lutheran representing Luther’s view so well. Luther and Calvin both believed that our assurance comes from our belief that Christ died for sin in general, and not faith that we are ‘elect’. Both Luther and Calvin believed that if we have faith in Christ then we are ‘elect’.
The other things to remember is that Arminian thought was really not that developed while Luther was writing and even while Calvin was writing minute detailed responses to Arminian beliefs, such as that found in John Owen’s ‘Death of Death, in the Death of Christ’ treatise was not needed.
In both Luther and Calvin, they limited atonement in scope. Calvin limited more strictly in eternity but neither limited the atonement in its power by making it dependent on human effort.
But to answer the question specifically, 'How did Luther think it worked?' He didn't he did not see much value in thinking too much about things he thought were incomprehensible.
Because Luther did not think about predestination that much some of the more interesting things he said were recorded by his close friends:
No. 3655b: Speculations About Predestination Unhealthy December 25, 1537
He [Martin Luther] spoke at length about the idle people who occupy themselves with disputation about predestination beyond the limits of Scripture. It is flue most ungodly and dangerous business to abandon the certain and revealed will of God in order to search into the hidden mysteries of God.” (Luther's Works 54.249)
No. 514: Predestination Is an Impenetrable Mystery Spring, 1533
“Paul wasn’t discussing predestination322 with the Jews but was disputing only with those who opposed it and said, ‘We are the people of God, we have the fathers, the promises were given to us,’ etc. He didn’t touch upon predestination except to repudiate the righteousness of the law. Paul said that we should not preach as if we could become good by our works, our fathers, etc. These don’t help. The others, those who believe, have it just as good as those who have fathers, etc. God has decreed it so from eternity. Hence Paul attacks only this.
“By grace alone are we saved. God doesn’t want to be obligated to anybody. Once we believe, he tells us (and this, too, is by grace), ‘Give, and it will be given to you [Luke 6:38]. You are bound to give in any ease, whether out of pity for the Turks or some other unfortunates, so you may just as well do it when I command it.’ We have no claim on him. So Esau and Jacob also testified to righteousness.323
“Otherwise one can’t settle the deep questions of theology, but Paul simply argues that it is by grace alone that we are saved. He meant to stick to this. Think what it means that you are not saved by the law, etc., but the Jews have works! Nobody but Christ can solve this. Hence he [Paul] said, ‘O the depth [of the riches and wisdom and knowledge of God,’ Rom. 11:33]!” (Luther's Works 54.90)