What do immersionists believe it means to be buried with Christ through Baptism? Romans 6:1-4 (NIV)
  
  What shall we say, then? Shall we go on sinning so that grace may
  increase? By no means! We are those who have died to sin; how can we
  live in it any longer? Or don’t you know that all of us who were
  baptized into Christ Jesus were baptized into his death? We were therefore buried with him through baptism into death in order that,
  just as Christ was raised from the dead through the glory of the
  Father, we too may live a new life.


How do immersionists/Baptists understand the meaning of being buried with Christ? What significant role does being buried with Christ have on the baptized person's spiritual life?

Do immersionists use this verse as support for Immersion Baptism?
I can only answer for my denomination (Baptist), but it was used as a supporting scripture for Immersion. During baptism, pastors would often say 


  Buried in the likeness of His death, raised in the likeness of His resurrection


or 


  ...raised to walk in newness of life


The idea is that baptism is a the believer's public confession of faith. And the immersion itself is a symbol of Christ's burial and resurrection.