What precisely does it mean to have the laws written upon our hearts?
It means that you have God's holy spirit, the spirit of Christ, indwelling. It means that so long as you let the spirit of Christ be expressed through you, that you will love God with all your heart, soul and mind and love your neighbor as yourself.
The apparent effect of this on you will be that you seem to be keeping the ten commandments. You conform your thoughts, heart and behavior to God's commandments because God's commandments define love:
Joh 15:10 If ye keep my commandments, ye shall abide in my love; even as I have kept my Father's commandments, and abide in his love.
1Jo 5:2 By this we know that we love the children of God, when we love God, and keep his commandments.
1Jo 5:3 For this is the love of God, that we keep his commandments: and his commandments are not grievous.
1Jo 5:4 For whatsoever is born of God overcometh the world: and this is the victory that overcometh the world, even our faith.
1Jo 2:3 And hereby we do know that we know him, if we keep his commandments.
1Jo 2:4 He that saith, I know him, and keepeth not his commandments, is a liar, and the truth is not in him.
1Jo 2:5 But whoso keepeth his word, in him verily is the love of God perfected: hereby know we that we are in him.
1Jo 2:6 He that saith he abideth in him ought himself also so to walk, even as he walked.
Rom 13:8 Owe no man any thing, but to love one another: for he that loveth another hath fulfilled the law.
Rom 13:9 For this, Thou shalt not commit adultery, Thou shalt not kill, Thou shalt not steal, Thou shalt not bear false witness, Thou shalt not covet; and if there be any other commandment, it is briefly comprehended in this saying, namely, Thou shalt love thy neighbor as thyself.
Rom 13:10 Love worketh no ill to his neighbor: therefore love is the fulfilling of the law.
More precisely, God's laws being written on our hearts is the process of overcoming, the process of Christian growth, through obedience to the indwelling of God's spirit:
2Co 10:5 Casting down imaginations, and every high thing that exalteth itself against the knowledge of God, and bringing into captivity every thought to the obedience of Christ;
Rom 6:16 Know ye not, that to whom ye yield yourselves servants to obey, his servants ye are to whom ye obey; whether of sin unto death, or of obedience unto righteousness?
In other words, in the Christian walk, we strive to put aside our own thoughts, ideas and behaviors and let Christ live instead:
Gal 2:20 I am crucified with Christ: nevertheless I live; yet not I, but Christ liveth in me: and the life which I now live in the flesh I live by the faith of the Son of God, who loved me, and gave himself for me.