This is one of the the key innovations of the Illumina (originally Solexa) sequencing.
This step is genius because sequencing is basically a signal-to-noise problem, as you are trying to observe individual tiny nucleotides mixed with billions of other tiny nucleotides.
With bridge amplification, we group some of the nucleotides together, and multiply the signal millions of times for that part of the DNA.
Figure 1. Illustration of the bridge amplification step of Illumina sequencing. Source.