What is solid-phase bridge amplification?
This stage is after the adsorption of the adapter-ligated fragments on the flow cells. 
The fragments are isothermally amplified to generate clusters [1]. The exact nature of the enzyme mix that catalyses this step is not disclosed by Illumina. I assume it must be similar to Helicase mediated isothermal amplification, which uses helicase and single strand-binding proteins (SSB) to denature the template. The polymerization is catalysed by a mesophilic DNA-Pol such as Klenow (E.coli).