The article you have linked to does explain everything thoroughly, though in a somewhat complex way. I did not know how much genetics you actually know, so I tried to answer very thoroughly.
So the article states that the gene Mendel studied and that was responsible for the white color of the pea flower is ANTHOCYANIN1 (in the article it is also called - bHLH, that is the name of the protein family that ANTHOCYANIN1 belongs to). So, ANTHOCYANIN1 is a transcription factor - a molecule that activates the transcription (making of mRNA) of other genes, basically it activates other genes.
The white flowered peas have a mutation (Guanine to Adenine transition) in the sixed intron of the gene ANTHOCYANIN1. Introns are sequences of non-coding DNA, that does not encode amino acids, and is removed from the RNA transcript by a process called splicing. The mutation occurs at a binding site for the Spliceosome (a protein and RNA molecule) that splices the RNA, because of this the RNA transcript is not spliced correctly. The faulty RNA transcript for the gene ANTHOCYANIN1 contains 8 extra nucleotides and because the genetic code is a triplet, a frame shift occurs. When a frame shift occurs all of the codons bellow the frame shift sight get changed. In this case, this leads to a premature stop codon. The translation of the ANTHOCYANIN1 RNA get stopped before a full protein can get synthesized and a non-functional protein is made.
If ANTHOCYANIN1 does not function, no genes are activated and no pigment is synthesized.
To summarize - the white pea flower color is a consequence of a nucleotide mutation (Guanine to Adenine) in a intron sequence of the gene ANTHOCYANIN1, this leads to incorrect splicing of the ANTHOCYANIN1 RNA (a frame shift) and to the production of a shorter protein that is non-functional.