"For if the dead are not raised, then Christ has not been raised either. And if Christ has not been raised, your faith is futile; you are still in your sins." NIV translation
Paul reminds the people that Jesus was crucified for our sins as the perfect sacrifice. The perfect sacrifice could not be a liar and Jesus had said that He would be raised from the dead. Therefore, if He had not been raised from the dead, it would be a sign of the rejection of the offering. Without His offering, there was no payment for our sins and so man would still be "in your sins".
"Then those also who have fallen asleep in Christ are lost. If only for this life we have hope in Christ, we are of all people most to be pitied." NIV translation
Have you ever been in a situation where you know someone is being deceived by another person and yet they will not listen to the truth? At that point, we feel sorry for them and that is what Paul describes here. The epicureans of Corinth were wallowing in their sin and loving it because they did not believe in an afterlife. If Christians (who did not take part in those parties) were wrong and died without an afterlife, then, they had just missed out on all that the world had offered.
"But Christ has indeed been raised from the dead, the firstfruits of those who have fallen asleep." NIV translation
Paul moves on by simply stating the fact that Jesus was raised from the dead. He knew that because he had seen Him and talked with Him.
The term "firstfruits" or "bikkurim" in Hebrew literally means "promise to come". As part of the Law that was given to Moses, the best of the land was to be brought to God during the three annual feasts. During their time in the wilderness, this offering was brought to God in faith that He was going to keep His promise to bring them to the land that he had promised. After they were living in the land, it was brought in thanks for what God had done in keeping His promise. These offerings were only the first part of the harvest that was standing in the fields.
The same principle applies with the resurrection of Christ as it is our "promise to come". He is the first and the best of what has already been provided by God.