In the Down syndrom typically, the translocation does not induce (usually; see later) any disease. We call someone carrying the translocation a "balanced carrier". The problem arises after, at the moment of segregation.
When balanced carriers reproduce
Consider someone who has a translocation as you showed. You showed only one chromosome of each types (1N), below are the 2 chromosomes (2N). On the below image, the person with the translocation is assumed to be mating with someone who does not have the translocation.
The balance carrier is usually healthy. As a consequence of this translocation a kid has probability 1/4 to be "normal", 1/4 to have the translocation (and therefore to eventually have offspring with trisomy/monosomy) and a probability 1/2 to have trisomy/monosomy.
Citing from wikipedia:
Most balanced translocation carriers are healthy and do not have any symptoms. But about 6% of them have a range of symptoms that may include autism, intellectual disability, or congenital anomalies. A gene disrupted or disregulated at the breakpoint of the translocation carrier is likely the cause of these symptoms.
Indeed, even a balanced carrier can be affected by the translocation. Gene regulation network is often quite complex and separating a gene from its regulatory region can eventually have important impact. It is also possible that the break point occurred in the middle of a gene (or a regulatory sequence).
I think that one cannot say much more about why a balanced carrier can be sick in a general form. We'll need to go case by case and ask specifically what are the genes and regulatory processes involved in the disease. However, it is easy to conceptualize that moving genes around can affect the genetic network and the expression of genes.