Leviticus 7:4

And the two kidneys, and the fat that is on them, which is by the flanks, and the caul that is above the liver, with the kidneys, it shall he take away:

The verse directs that the kidneys with their flank fat and the covering over the liver were to be removed as part of the trespass offering, which is most holy and belongs to the ceremonial law given because of sin.
The guilty person carried his iniquity until he brought and slew the offering in faith in the coming Redeemer, when the priest made atonement with its blood and transferred the sin in figure to the sanctuary, where the forgiven sin remained until it was put away on the tenth day of the seventh month.
The fat of the kidneys was burned as holy upon the altar, while the flesh was burned outside the camp as unholy.
Like the other details of the sacrificial laws, this instruction finds its fuller and symbolic meaning in the sacrifice, life, and priestly ministry of Jesus Christ in the heavenly sanctuary, showing God's hatred of sin and that sin demands a penalty, which Christ met when He gave Himself as an offering and bore human iniquities.
The fat, understood as oil, represents the grace of the Holy Spirit given through Christ's life, and even in the poorest sinner and most repeated trespasser some of that Spirit-given likeness of God remains as holy to be renewed after God's image rather than destroyed, unless the Spirit is wholly grieved away so that no desire to repent remains.

Ritual Detail and Setting

The two kidneys were to be taken away, with the fat upon them by the flanks, and the caul above the liver with the kidneys. These directions belong to the regulations for the trespass offering. That offering is classified as most holy.

The same context is connected with the rule that the flesh of peace offerings was eaten. The instruction belongs to the ceremonial law, distinct from the moral law, as a law instituted in consequence of sin. It stands among the instructions for the particular individual offerings brought during the year, which formed part of the priests' daily ministry in the first apartment of the earthly sanctuary in the yearly round of service.

A person bore his iniquity until he brought and slaughtered the offering, and the priest then made atonement with its blood and the person was forgiven and free from that iniquity. The victim's blood, sprinkled by the priest before the veil and on the altar, was the means by which the iniquity was transferred to the sanctuary. The individual offerings described in the first seven chapters of Leviticus were brought in faith by people burdened with sin and expressed faith in the coming Redeemer. Together with the daily offering they transferred in figure, through the blood of the sacrifice or the person of the priest, the sins of the penitent to the sanctuary, where, the blood having been sprinkled on the altar before the ark, the forgiven sins remained through the year until they were cleansed and put away on the tenth day of the seventh month.

The fat associated with the kidneys in the sin offering and trespass offering was removed and burned as holy on the altar. The flesh, by contrast, was burned outside the camp as unholy.

Fulfillment in Christ's Work

Every detail in the laws for sacrifices and offerings had meaning, and the fuller and symbolic meaning of this instruction concerning the different kinds of sacrifices is found in the sacrifice, life, and work of Jesus Christ.
The sanctuary service, including these instructions and its yearly round of service, served as a type of Christ's priestly ministry in the heavenly sanctuary and taught about God's way and Christ's work for humanity.
The offerings in Leviticus 1-7 are shadows of their fulfillment in Jesus Christ and show God's hatred of sin and that sin demands a penalty.
These offering instructions stand alongside the truth that Christ gave himself as an offering for sin and bore people's iniquities.

The fat, also understood as oil, represents the grace of the Holy Spirit, since oil was used in anointing as a sign of the Spirit's gift and the Spirit's grace is the result of Christ's life in people.
Even in the poorest sinner and the most repeated trespasser some of this Spirit-given likeness of God remains, and it is holy and is to be renewed after God's image, not crucified.
But if the Spirit is wholly grieved away and divine light wholly becomes darkness, the sinner has no desire to repent or hunger after God.