SSRIs are speculated to work via activation and downregulation of certain serotonin receptors, in particular 5HT1A, an autoreceptor responsible for regulating release of other neurotransmitters in the brain. Buspirone acts directly on 5HT1A as an agonist, I believe, which is more direct than the shotgun approach of SSRIs. In theory, activation of 5HT1A and its downregulation can help with depression and anxiety. Buspirone also blocks some dopamine receptors, but not any that are associated with dopamine blockade from antipsychotics, so their serious dopamine-related side effects aren't something to worry about, but the blocking might also contribute to anxiolytic effects.

Newer SSRI-like drugs that block the serotonin transporter and activate 5HT1A also exist, combining the actions of both drugs.

It typically takes serotonin receptors about two weeks to downregulate, and possibly even months to see any effects on depression and anxiety. Buspirone is not like benzodiazepines that work pretty much instantly, but it doesn't come with any of the risks benzodiazepines do.