They prevent reuptake and termination of action of serotonin at the synapse. The enhanced availability of serotonin leads to improved serotonergic neurotransmission; this in turn helps to improve parameters such as mood e.t.c., hence, their use as antidepressants.

The inhibition of reuptake of serotonin is only part of the story. While the reuptake of serotonin by SSRIs works immediately, the antidepressant effect of SSRI does not happen immediately. In rats SSRIs slow the firing of serotonin neurons due to the action of serotonin on presynaptic serotonin receptors. However, after 2 weeks of administration the firing rate of serotonin neurons returns to normal, due to desensitization of presynaptic serotonin receptors. This is one theory of the delay in antidepressant action of SSRIs and is reviewed in detail here http://pharmrev.aspetjournals.org/content/51/3/533.long. Another theory is that antidepressants induce neurogenesis and that is responsible for the antidepressant effect https://www.ncbi.nlm.nih.gov/pubmed/25455365.

I agree with Dr. Young, and I would add that there are several molecular targets to which these SSRIs bind besides uptake transporters.  See: https://www.psychiatrist.com/JCP/article/Pages/1998/v59n07/v59n0701.aspx

for "not so selective SSRIs."  I should also add that in some invertebrates, for example C. elegans, the SSRI fluoxetine evokes responses in worms that lack serotonin transporters and in fact fluoxetine binds to G-protein coupled serotonin receptors.

I am convinced that the psychological effects of SSRIs are predominantly determined by their causing an increase in extracellular serotonin and that this increase is immediate and relatively constant over the time that clinical doses are administered. Thus, in primates, levels of cisternal serotonin increase ~3-fold within hours of administration and remain at that level over a month-long administration period (https://www.ncbi.nlm.nih.gov/pubmed/15452685). Also, in humans, effects of SSRIs on perception of negative scenes and emotions can become more positive within an hour of administration (https://www.ncbi.nlm.nih.gov/pubmed/15229059) . Finally, meta-analysis has shown that the antidepressant effects can be discerned within the first week of administration (https://www.ncbi.nlm.nih.gov/pubmed/17088502) . So, I think the idea of a delay in response, both neurobiologically and psychologically, has been overstated and is less than useful.