He is referring to one's unwillingness to follow him. Someone who wants to save their life is someone who wants to continue indulging their sinful desires, living as they please, and ignoring God. This refusal to let go of sin will result in the loss of the eternal life of the soul. Man can enjoy every earthly delight but in doing so, his soul is doomed. He will not have eternal life with God. Jesus teaches us that life continues on into eternity. If you deny Christ in an attempt to save physical life, you will lose eternal life and salvation.1
35 For whoever would save his lifewill lose it, but whoever loses his life for my sake and the gospel’s will save it.