It is giving up his own life, to take on his mother' s for her sake. The sentiment of the poem is correct. For Sutskever to take up his mother's life would compound the tragedy. If he wants to honor her, it is enough for him to live. It is that which ensures her immortality.
Life is complete and each life is unique.
To give up life, to deny one's own specialness is to commit a sin. She tells him that the seed contains the flower, the acorn the tree. She will remain alive because he exists. His very existence attests to hers. In this way her presence is real and eternal.