The speaker is looking after an aging mother, lifting her up and down the stairs the way a parent once lifted a small child. The roles have reversed, and the poem says so plainly in its first line. As he carries her, he talks about ordinary things, like the importance of wearing good clothes, which feels like the kind of fussing a parent might do for a child. It is a gentle, slightly tender kind of small talk.
The turn comes in the third stanza. The speaker says he trusts his hands "because of their mortality." Hands that will age and fail like hers feel honest to him. The last lines go further: she gave him those hands, and in doing so gave him himself. The odd phrase "giving me me" makes the point simply, as if there were no fancier way to say it.
The mood is quiet, humble, and loving. There is sadness in the idea of mortality, but mostly the poem feels grateful.