Grounding is a electrical term.
They are suggesting that the negative - DC current/power of your motherboard goes thought the standoffs and grounds to your case. I don't think this is correct because if you have a acrylic case this would be impossible. But, You definitely want to install the stand-offs to keep your mobo from shorting out on the case.
It sounds like your PSU is no good, I would install the old PSU in the new case and double check all the connections.
One of my old cases wouldn't let the pci cards go all the way into the mobo until I took it apart and adjusted the case with a hammer