First of all, it's important to note that the staircase is an allegory for the mother's life. An allegory is a literary device which is used to reveal a hidden message, meaning, or moral. In this case, the speaker's mother uses the allegory of climbing a staircase to tell her son about what kind of life she's had and how she wants him to keep going in his, however many problems he may face.

As she explains, her life hasn't been easy; it's been "no crystal stair." On the contrary, it's been full of tacks and splinters, with boards torn up and places with no carpet on the floor. These literal obstacles to climbing the stairs represent the many challenges that the speaker's mother has had to face in her life.

Despite everything, though, she's just kept on climbing the stairs, reaching landings and turning corners, sometimes even in the dark. Having kept on going, she's keen for her son to do the same. Returning to the allegory, she doesn't want him to give up climbing and sit on the stair because it's too hard. She's still going, and she's still climbing, despite all the many obstacles in her path, and she hopes that her son will follow her example.