As far as I know, there's two meanings.
The first meaning.
Her big hand holds mine shows that the grandmother's hands are bigger than the persona's due to the persona still being a child.
White hand in black hand shows that the grandmother's hand is darker than the persona's because old people tend to have darker skin than young people.
The second meaning.
The grandmother's bigger hand represents that the grandmother has done a lot of thing. The hand being big may indicate that she had done big changes. The persona's hand is small because the persona haven't yet done anything much due to her being three and all that.
The white hand represents the persona's youth, purity, and innocence as a child. The black hand represents that the grandmother is experienced, and an old soul.