Walking in the spirit means living life to please God rather than to please yourself or others. It means obedience to God rather than obeying your own needs and desires, or the needs and desires of others.
This is the way Jesus lived his life.
John 5:30.
I can of mine own self do nothing: as I hear, I judge: and my judgment is just; because I seek not mine own will, but the will of the Father which hath sent me.
John 8:28-29.
Then said Jesus unto them, When ye shall have lifted up the Son of man, then shall ye know that I am [he], and [that] I do nothing of myself; but as the Father hath taught me, I speak these things.
And he that sent me is with me: the Father hath not left me alone, for I do always those things that please him.
This is also the way he expects those who follow him to live their lives.
Mark 3:35.
For whosoever shall do the will of God, the same is my brother, and my sister, and mother.
There are two ways a person can live life. The first way is to live to please yourself.
Ephesians 2:1-3.
And you [hath he quickened], who were dead in trespasses and sins;
Wherein in time past ye walked according to the course of this world, according to the prince of the power of the air, the spirit that now worketh in the children of disobedience:
Among whom also we all had our conversation {or behavior} in times past in the lusts of the flesh, fulfilling the desires of the flesh and of the mind; and were by nature the children of wrath, even as others.
I John 2:15-17.
Love not the world, neither the things [that are] in the world. If any man love the world, the love of the Father is not in him.
For all that [is] in the world, the lust of the flesh, the lust of the eyes, and the pride of life, is not of the Father, but is of the world.
And the world passeth away, and the lust thereof: but he that doeth the will of God abideth for ever.
Walking in the spirit is God's plan for his people to overcome sin and live a life that is pleasing to Him.
Galatians 5:16-18.
[This] I say then, Walk in the Spirit, and ye shall not fulfil the lust of the flesh.
For the flesh lusteth against the Spirit, and the Spirit against the flesh: and these are contrary the one to the other: so that ye cannot do the things that ye would.
But if ye be led by the Spirit, ye are not under the law.
There are two major keys to walking in the spirit.
The first is developing a Jesus frame of mind regarding God, the world, self, others, and all that the world has to offer. This is done by immersing onesself in God's Word, and allowing it to change the way you think. This is called putting on the mind of Christ, renewing your mind, setting your thoughts on things that are above, and minding the things of the spirit.
The second is developing a habit of obedience to God, even when you do not feel like doing what He says do. Doing what God says is proof that you love Him. If you hear God's Word but don't do it, you are deceiving yourself.