"that Christ died for our sins in accordance with the scriptures" (15:3)
"and that he was buried, and that he was raised on the third day in accordance with the scriptures" (15:4).
We can, of course, head off in a wonderful and (it may turn out, re the resurrection) painstaking search for the texts which signify what Paul means by "in accordance" with the scriptures of Israel. As we do so we might speculate whether Paul was talking in a general sense about Christian reading of the scriptures of Israel or in a particular sense about circulating lists of texts which Christians were treasuring as underpinning their understanding of Christ's death and resurrection.
More briefly, we can note the confidence Paul expresses, likely reciting a familiar creedal statement, that Christ's dying "for our sins" and Christ's resurrection were anticipated in the scriptures. There was presumably a point before his conversion (Acts 9) where he viewed Jesus' death as an unfortunate early end to life and doubted if not denied reports received about Jesus' followers which claimed he had risen from the dead. Now Paul re-reads familiar scriptures and finds some make sense of the significance of Jesus' death, that death being "for our sins" and other texts forecast Jesus' resurrection.
Combining an understanding that Jesus was always going to be raised from the dead with the evidence of his appearances to his followers and to his family (15:5-7), Paul is in no doubt that when Jesus "appeared also to me" (15:8), the appearance was not an apparition of a man now dead and buried but an appearance of one who once was dead and once was buried but now has been raised to life.
That confidence that Jesus had been raised from the dead then drives forward the rest of 1 Corinthians 15 as Paul battles some specific opposition over the idea of "resurrection of the dead," an idea which is being denied by some of the Corinthians.
Paul places a theological "lock" on the connection between Jesus' death and Jesus' resurrection:
"If Christ has not been raised, your faith is futile and you are still in your sins." (15:17)
If Christ was not raised from the dead then he did not die for our sins.
The resurrection is not an "add on extra" or a "bonus" to the story of Jesus living for God and dying a death which demonstrated his faithful obedience to God's will for his life. The resurrection is the sign (we might say THE SIGN) that Jesus was unique among many faithful Jews of his day.
He was in a class of his own as a faithful Jew: the Anointed of God, the suffering servant who would suffer for the sake of the world, the Paschal Lamb of God who would die for the redemption of the world. Deny the resurrection and Jesus is not only dead and still buried but his death is without import or effect on the lives of any other people.
Conversely, if we accept the Christian proclamation that Jesus was raised from the dead, then we cannot deny that he died for our sins. When we rejoice in the resurrection we should not only rejoice that Jesus overcame death, we should rejoice that our sins are forgiven.
In this way, new creation comes into being, with Christ as the new Adam (Primal Human) of the new creation (1 Corinthians 15:21-22): sins forgiven means the Fall is undone, death gives way to resurrection, sin gives way to righteousness, (evil) dominion/power/authority is abolished (15:21-28). The past is healed, the present anticipates the future, the future is the full harvest of which Christ is the firstfruits (15:23).
None of this is communicated by the empty tomb or the appearances (as bare facts of history); all of it flows from understanding them "in accordance with" the scriptures of Israel through the Holy Spirit that simultaneously births the new scriptures of the church as Paul and others write that understanding down.