Unity in the Body of Christ comes due to the fact that Christ baptizes each believer with the Holy Spirit into his spiritual body. This is true for all Christians from the moment of their salvation and forevermore.
And if this is difficult for us to wrap our minds around, Paul gives us one more picture at the end of the verse to help us put a sort of concreteness to what is going on here—“and all were made to drink of one Spirit.” Here again the Spirit is like water, but Paul changes the image from one of immersion in water to one of drinking water. The Spirit is not only the means through which Christ unites all of us into one body, the Spirit indwells us like when we drink a glass of water, and there is no exception. Despite our diversity, every Christian was made to drink of one Spirit—every Christian has the Holy Spirit dwelling within him.
And Paul makes the supernatural nature of this explicit right there in the verse by reminding us of the diversity that exists, not just the functional diversity of Spirit giftedness within the body, which is his main point in the passage, but diversity that really should to a greater degree prevent our unity. “For with one Spirit we were all baptized into one body—Jews or Greeks”—these are the two extreme representative polls of ethnic diversity—“slaves or free”—the two extreme polls of social diversity; you can’t get any more diverse socially than the difference between someone who is a slave and someone who is free. But as a Christian, these do not define you! In the providence of God this is who you are, but this is not where you find your ultimate identity. It does not matter what your ethnicity is, whether you are a Jew or Greek, black or white, Asian or Germanic, if you are a Christian, then your identity is one who has been immersed with one Spirit into one body. It does not matter what your vocation or social status is, whether you are a slave or free, a business owner or a plumber, an accountant or a professor, wealthy or poor, if you are a Christian, then your identity is one who has been made to drink of one Spirit. And if these diverse characteristics do not define individuals within the body or divide the body, as they naturally would tend to do, certainly functional diversity should not divide the body. It does not matter what your interests are, what your gifts are, what your personality is, whether you are introverted or extroverted, whether you enjoy public speaking or would rather serve quietly in the background, whether you have musical talents or your gifts are geared toward quiet encouragement, despite all of that diversity, we are one in Christ! This is not natural; this is supernatural.