In order to analyze the poem, a good place to begin is by figuring out what you think its point is. What do you believe the poet wants us to understand? I think Millay attempts to redefine childhood, suggesting that it isn't about how old one is but, rather, is about whether or not one has lost a person who truly matters to them to death: someone really crucial to one's life, like a parent.

It sounds as though the speaker has watched one of their parents decline—their "self" already gone. The speaker tried, perhaps, to tempt them with raspberry jam, to cajole them back to life, but it didn't work. This experience, the speaker seems to claim, is what ends one's childhood.

This poem begins with a metaphor comparing childhood to a place where nobody dies. In the innocence of childhood, nobody really does die, the author says. Oh, sure, old people die, and maybe your cat will die, but not important people - not people that matter - certainly not your parents.

The visual image I get from reading this poem is that the author is a grown woman sitting at the table with her elderly mother, drinking tea:

To be grown up is to sit at the table with people who have died,
who neither listen nor speak;
Who do not drink their tea, though they always said
Tea was such a comfort

These lines illustrate that the speaker is now grown up, but she is thinking back about how easier it was as a child, when she didn't realize that people DO die, even parents. Now, she is sitting at the table with an older person who appears to be rather senile, who does not pay attention to her tea -- who does not really pay attention to anything:

Shout at them, get red in the face, rise,
Drag them up out of their chairs by their stiff shoulders and shake
them and yell at them;
They are not startled, they are not even embarrassed; they slide
back into their chairs.

The speaker seems to be frustrated with the elderly person - her mother, perhaps - and longing for a time when she did not have to face the reality that her mother is old and won't be around for long.

Does this help? Go over the poem and look for the language that creates a melancholy mood and see if you agree with me.