How do signal transduction pathways utilize transcription factors to express a specific gene?
This is a combination of multiple regulatory systems. Most genes are not regulated by a single factor, but by many. Moreover in eukaryotic organisms there is also epigenetic, which "inactivates" permanently certain areas of the genome by compacting these zones forming heterochromatin. 

Moreover, even when we are interested in a single function a TF might have, that does not mean that it only has this function in the cell. We view it as humans, and maybe for us it is logical that a response to a high metabolite in the environment simply leads to expressing the protein that metabolizes it. However, probably it is interesting for the cell to also activate other pathways and gene networks (i.e. anabolic pathways that feed off this metabolite, compensatory pathways to mantain homeostasis...). Because that normally TFs affect more than one gene naturally. 

The mechanisms of action of the TFs are multiple, but they tend to simply allow or deny the access to the adequate RNA polymerase. For more information about this check the wikipedia article on TF structure  

In this image we see a typical mechanism of action, DNA bending 

Different domains interact in activating a gene, enhancers which are usually long distance, promoters which tend to be in the upstream region, though they can be downstream or in the gene sequence, and epigenetic factors that change the compaction of DNA, making it inaccessible to proteins. 

Just as a last thing to point out, you should acknowledge that the biological systems are far from exact and yes, a single TF might activate 200 genes, even when its main objective in that moment is to activate one, but it won't really matter as long as the those 199 activated genes express at a very low rate (for example by not having the right RNA polymerase subunit accessible, being marked as inactive by epigenetic systems or having their own inhibitors).