Transcription and translation of prokaryotic operons
As you pointed out, the repressor gene lacI is transcribed as a one mRNA, and three structural genes: lacZ, lacY and lacA are transcribed into a single polycistronic mRNA. The two mRNAs are translated independently of one another.

The polycisronic mRNA is not broken into pieces. Rather, it is translated by ribosomes (at least three, explanation below), giving rise to three individual proteins.



How from one mRNA three protein products are translated? The polycistronic DNA consists of a promotor (Plac), which ensures that transcription will be initiated, and three open reading frames (ORFs) for for lacZ, lacY, and lacA. Upstream of each of the ORFs, ribosome binding sites (RBSs) are located. Translation is initiated by the binding of the ribosome with both the RBS and the start codon. Thus, multiple ribosomes can bind upstream of the three ORFs and thus resulting into three proteins.

There is not very thorough literature on the translation of polycistronic RNA in prokaryotes, mainly because the condition which is necessary for translation is the RBS and there is nothing specific for polycistronic translation. A brief overview on the subject can be found in several places, as I recommend the following:


Ralston, A. (2008) Operons and prokaryotic gene regulation. Nature Education
From Genes to Genomes: Concepts and Applications of DNA Technology 
Molecular cell biology
Analysis of Genes and Genomes