"A Broken Appointment" begins with its speaker clearly feeling let down. He had an "appointment"—that is, a meeting/date—with a woman, but she stood him up: "You did not come," the speaker says bluntly, addressing the woman directly. Given that the woman isn't around to respond, this is an example of apostrophe. The brevity of this opening line captures the speaker's shock and hurt at the woman's decision not to come.
The speaker clearly has strong feelings for this woman, and, in line 2, he describes coming to the crushing realization that she wasn't ever going to show. "Time" cruelly marched on; the hour of their "appointment came and went, and the clock kept ticking, indifferent to the speaker's emotions. The speaker grew "numb" with waiting, his emotions worn thin.
While the opening line contained just four blunt syllables, the second line falls into iambic pentameter: a meter in which each line contains five iambs, poetic feet with an unstressed-stressed syllable pattern (i.e., da-DUM).
And march- | ing Time | drew on, | and wore | me numb,—
The longer length of this line helps to convey the speaker's discomfort. The line stretches on, capturing how time seemed to "draw on" while the speaker waited. The regular iambs also seem to "march" like time: left, right, left, right; da-DUM, da-DUM, da-DUM.
Finally, these opening lines form a rhyming couplet: "come"/"numb." This perfect rhyme lends the opening a sense of firmness and finality. The woman didn't show, and there's nothing the speaker can do about it.