That is G-d's command. It is still in practice today with "Shatnez Labs" all over the world.
WHAT'S THE REASON?
The Torah does not explain the reason for shatnez, and it is categorized as a chok -- a law whose logic is not evident. The Torah has many such laws; we do not know why pork is forbidden, for example. And the prohibition of shatnez is equally strong.
Why did God make a chok in the first place? What's the purpose of a commandment whose reason we have no inkling of?
The power of a chock is as follows: If the reasons for all the mitzvot were as obvious as "don't murder" or "don't steal," then a person could go through life without developing a relationship with God. How so? Just as there are many fine, upstanding people who don't murder -- not because they believe in God, but simply because they understand that it's wrong -- we might likewise observe mitzvot simply because they "make sense."
Leaving God out of the picture would be missing the point entirely.
Leaving God out of the picture would be missing the point entirely. That would be humanism, not Judaism.
Having said all this, God still wants us to use our intellect to understand the mitzvot to the best of our ability. Thus the commentators suggest different "explanations" for shatnez.
One idea is that he mixing wool and linen upsets the environmental and/or metaphysical fabric of the universe. God created different species that work together in the symphony of creation. Our job is to respect and appreciate this diversity and help maintain this special orderliness.
The Midrash suggests that the reason stems from the story of Cain and Abel, as recorded in Genesis chapter 4. Cain brought God an offering of flax (the source of linen) and Abel brought a sheep (wool). The incident resulted in Cain killing Abel, and it was thus decreed that never again shall the two substances mix.
This is perhaps hinted to by the Torah juxtaposing the prohibition of shatnez with the imperative to "love your neighbor as yourself" (Leviticus 19:18-19). Each person must cherish his own uniqueness and not feel threatened by others. Cain did not understand that he and his brother had different tasks in life, different roles in creation.