This element of righteousness comes with maturity.  The more we address all the elements of righteousness, the more this element manifests itself.  Notice this in the commentaries for both verses.

Phil 4:7

[And the peace of God] The peace which God gives. The peace here particularly referred to is that which is felt when we have no anxious care about the supply of our needs, and when we go confidently and commit everything into the hands of God. "Thou wilt keep him in perfect peace whose mind is stayed on thee;" Isa 26:3.

[Which passeth all understanding] That is, which surpasses all that people had conceived or imagined. The expression is one that denotes that the peace imparted is of the highest possible kind. The apostle Paul frequently used terms which had somewhat of a hyperbolical cast and the language here is that which one would use who designed to speak of that which was of the highest order. The Christian, committing his way to God, and feeling that he will order all things aright, has a peace which is nowhere else known. Nothing else will furnish it but [true] religion. No confidence that a man can have in his own powers; no reliance which he can repose on his own plans or on the promises or fidelity of his fellow-men, and no calculations which he can make on the course of events, can impart such peace to the soul as simple confidence in God.

[Shall keep your hearts and minds] That is, shall keep them from anxiety and agitation. The idea is, that by thus making our requests known to God, and going to him in view of all our trials and wants, the mind would be preserved from distressing anxiety. The way to find peace, and to have the heart kept from trouble, is thus to go and spread out all before the Lord; compare Isa 26:3-4,20; 37:1-7. The word rendered here "shall keep," is a military term, and means that the mind would be guarded as a camp or castle is. It would be preserved from the intrusion of anxious fears and alarms.

[Through Christ Jesus] By his agency, or intervention. It is only in him that the mind can be preserved in peace. It is not by mere confidence in God, or by mere prayer, but it is by confidence in God as he is revealed through the Redeemer, and by faith in him. Paul never lost sight of the truth that all the security and happiness of a believer were to be traced to the Saviour.    ~ Barnes

Note:  Peace is a fruit of the Holy Spirit in us, for us.  Mankind does not know nor understand this kind of peace.  This is the highest order of peace, the kind that Jesus had on earth, as a human like us.  Like Jesus, we are to take and give all our troubles and worries to God.  We are to guard our minds and not allow negative thoughts to dwell there.  This is only possible if Christ dwells in us.

Now the commentary on Col 3:15:

[Rule in your hearts] Brabeuetoo (NT:1018). Let the peace of Christ judge, decide, and govern in your hearts, as the brabeus, or judge, does in the Olympic contests. No heart is right with God where the peace of Christ does not rule; and the continual prevalence of the peace of Christ is the decisive proof that the heart is right with God. When a man loses his peace, it is an awful proof that he has lost something else; that he has given way to evil, and grieved the Spirit of God. While peace rules, all is safe.  ~ Clarke

Col 3:15-16

The peace of Christ (RSV). That peace which Christ mediates to those in union with him (cf. John 14:27; Rom 5:1). It is to rule in the sense of arbitrating differences that arise in the body (Bruce). Similarly, the indwelling word of Christ, i.e., his teaching, exercises a transforming influence on a believer's life. ~ Wycliffe

Note:  We are transformed by the Word of God.  Notice from Word of God…
 

Peace I leave with you, my peace I give unto you: not as the world giveth, give I unto you. Let not your heart be troubled, neither let it be afraid.  –John 14:27

Therefore being justified by faith, we have peace with God through our Lord Jesus Christ. –Rom 5:1

The LORD will give strength unto his people; the LORD will bless his people with peace. –Ps 29:11

For unto us a child is born, unto us a son is given: and the government shall be upon his shoulder: and his name shall be called Wonderful, Counseller, The mighty God, The everlasting Father, The Prince of Peace. –Isa 9:6

These things I have spoken unto you, that in me ye might have peace. In the world ye shall have tribulation: but be of good cheer; I have overcome the world. –John 16:33

For to be carnally minded is death; but to be spiritually minded is life and peace. –Rom 8:6

For he is our peace, who hath made both one, and hath broken down the middle wall of partition between us.

–Eph 2:14

Lesson 28: Instructions – Let the peace of God rule in your heart. 
 
 1] Understand that this element of righteousness comes with spiritual maturity.
 2] Commit everything into the hands of God.
 3] Invoke peace [a fruit of the Holy Spirit] in every situation of your life.
 4] Take your troubles, trials and wants to God.
 5] Guard your mind against all negative input.
 6] Never lose sight of the fact that all peace, security and happiness is the work of Christ in us.
 7] Immerse yourself in the Word of God and be transformed by it.

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