The book of Leviticus is a key to understanding the death of Jesus. He died as the perfect sacrifice to atone for all sins of all people. His death fulfilled all the sacrificial order and he was the ultimate Day of Atonement (cf. Heb 9-10). Since his sacrifice was once for all time, no more sacrifices are needed.
The cultic laws inform Christians of the character of Yahweh's holiness. It is important to understand the principles underlying these laws so that Christians may see how they are to live among and yet be distinct from their neighbors. For example, the laws against offering children to Molech may address the contemporary issue of abortion. On the other hand, the fact that Jesus did away with the laws of ritual purity is evidence of the glorious freedom we have in the redemptive work of Jesus.
The moral laws speak loudly to Christians, for they are grounded in the revelation of Yahweh's character. Jesus took up the commandment to love one's neighbor as oneself (19:18) and declared it to be the second greatest commandment (Mt 22:39; Mk 12:31; Lk 10:27). Other laws like not reaping up to the border of one's field nor gathering the gleanings (19:9) have to be transposed in order to be applied to contemporary culture. But the principle of not hoarding all that one makes for oneself and sharing one's abundance with the poor and unfortunate is clear. These moral laws offer numerous examples and illustrations of how to apply the great law of love in everyday situations.
The parenthetic material inspires Christians as it motivated the Israelites to be careful to keep the Word of God. These exhortations carry the promise of life, i.e., a blessed, peaceful, long life in the land of promise (18:5). Jesus promises that those who believe in him have eternal life (Jn 6:47). This parenthetic material, on the other hand, solemnly warns of devastating punishment to those who do not keep Yahweh's statutes (18:24-30). So, too, Jesus warns of eternal punishment to all those who fail to trust in him (cf. Mt 25:41-46). As the song says, “Trust and obey, for there is no other way to be happy in Jesus.”