Without going into Down syndrome specifically, let's address the title of your question: Why is trisomy a problem? Or, more specifically, why can it be a problem to have extra copies of genes?
Let's imagine biology to be just a really complicated cake. If you are trying to bake a cake, and you leave out the flour, I think mostly everyone would agree that the outcome product wouldn't be much of a cake at all: that's the condition where you are completely lacking an ingredient, both copies of the same chromosome, a nearly assured fatality.
But let's say you instead include 50% too much flour. What results? You might still get a cake, but it might not taste perfect: maybe a little too bread-like or not sweet enough.
In short: having an extra chromosome means genes on that chromosome are overexpressed. Those gene products could themselves have deleterious effects in excess, or they could affect expression of other genes and cause over- or under-production of those products. Of course it isn't as simple as "+50% expression!" because of regulatory mechanisms ubiquitous in biology, but there is sufficient overexpression to cause some issues.
Really, someone with Down syndrome develops almost completely normally - many other trisomys are fatal in the early stages of development, so we don't become familiar with them. For Down syndrome in particular, you can check out some of the proposed mechanisms on Wikipedia. One suggestion is that some of the cognitive deficits resemble Alzheimer's disease, implying a causative function for amyloid beta in both circumstances. The hypothesis is that amyloid beta accumulates in aging brains, but people with Down syndrome overproduce amyloid beta so they show dementia symptoms much earlier. This comparison is certainly not the only factor in Down syndrome, however.