bicarbonate/carbonate is an acid/base couple therefore it's a buffer solution when pH is within one unit of the pKa (10.3). It's role in ELISA is to keep a stable pH around 10. Just like any other buffer, it is made of the mixture of the acidic form (bicarbonate) and the basic form (carbonate).
A tradition if you ask me. From a fundamental point of view, when adsorbing proteins to a hydrophobic surface (such as the wells of any elisa plate), to favor adsorption the protein should be at its pI, to "destabilize it" and favor the hydrophobic effect. Nevertheless, the hydrophobic effect is thermodynamically favored and the protein at any pH will be adsorbed to the wells surface to a extent good enough for elisa.
It is called carbonate/bicarbonate since you have to mix both solutions to prepare a buffer at pH 10. At pH 10.32 you will have half the molecules as carbonate and half as bicarbonate, hence the name.
I think that the relatively high pH of the bicarbonate/carbonate buffer that is often used for coating ELISA plates destabilizes protein structure sufficiently to expose the hydrophobic interior, which makes the protein more prone to sticking to the hydrophobic surface of the plate.
Polystyrene microplates are a type of matrix commonly used for immobilization of proteins. Proteins passively adsorb to the polystyrene surface through hydrophobic interactions. Generally, this adsorption of proteins onto the polystyrene surface occurs best in carbonate/bicarbonate buffer at an alkaline pH (9.0–9.5).