How does identifying the speaker and the occasion of the poem reveal the dramatic quality of the poem?

Certainly, the structure of a poem, such as a sonnet, an ode, or an elegy, contributes to its dramatic effect.  Then, the occasion of the poem and the tone of the speaker contributes to the dramatic effect.  In Thomas Gray's "Elegy in a Country Churchyard," for instance, there is the sobering tone of the Latin motif of momento mori, "Remember that you must die." And, as the speaker traverses the graveyard and reads the unknown names on the markers, he reflects that the lives of these country people may have been impeded from greatness simply because of their humble stations in life.  He also realizes that now they are the equal of any, as all humans reach the same end.  In fact, there are, perhaps, more noble because they never compromised their values in order to achieve greatness:

Far from the madding crowd's ignoble strife/Their sober wishes never learned to stray/Along the cool, sequestered vale of life/They kept the noiseless tenor of their way. 

Indeed, the structure of Gray's poem lends dramatic qualities to the lines, and the tone of this poem reinforces these qualities.