How can a chromosome translocation in somatic cells lead to disease?
You have made an incorrect assumption-- that all the genes on both chromosomes necessarily remain intact. It's possible that they do, and the the translocation is harmless.

But the transition point could be within a gene. This would usually lead to that copy of the gene becoming inactive, which can cause diseases just like any other loss-of-function mutation.

Alternately, the translocation can fuse two genes in a pathogenic way. While this might seem fantastically unlikely, the fact that the translocations are non-random means that there are patterns that recur frequently. The BRC-ABL fusion (https://en.wikipedia.org/wiki/Philadelphia_chromosome) is one of the most common causes of childhood leukemia. The fusion protein is constitutively active because the regulatory domain of the original protein has been replaced.

Finally, a translocation could separate a promoter element from the gene it controls, which could cause aberrant expression of that gene. The promoter element could also wind up proximal to a different gene and cause aberrant expression of that gene.