The lacZ gene is used as a selectable marker in several plasmids. It differentiates recombinants from non-recombinants based on their ability to produce colour in the presence of a chromogenic substrate.
The lacZ gene encodes a protein enzyme known as β-galactosidase. In the presence of a chromogenic substrate such as X-gal, it produces blue coloured colonies.
When a foreign DNA or gene of interest is inserted within the lacZ gene, the gene gets inactivated and fails to produce β-galactosidase (insertional inactivation). The presence of an insert in the plasmid, thus, results in colonies that do not produce any colour (colourless/white).
The colonies of bacteria that are blue in colour and lack the insert are identified as non-recombinant colonies and the bacterial colonies that are white and contain the insert within the plasmid are identified as recombinant colonies. Since this method relies on blue and white colours of the colonies for selecting recombinants, it is referred to as blue-white screening.
Since in the question that only a few of the bacteria took up the plasmid while many didn’t, the number of white colonies (recombinants) would be less than the blue colonies (non-recombinants).