how does the poem switch from person to person or between different times or places?

One obvious and concrete way that the poem switches slightly between topics and times is by starting a new stanza. Stanza 1 is being narrated by an unknown narrator. We don't know the narrator's gender, but we do know that they have an Aunt Jennifer that has some tigers that prance across a screen. The rest of the stanza stays fairly focused on the tigers. We get a brief bit about men beneath a tree, but then the next line returns to the tigers and their pacing.
Stanza 2 then shifts to Aunt Jennifer herself. We are told that Aunt Jennifer is struggling with her needlework, and that is likely because of her age. We also find out that Aunt Jennifer is/was married, and the marriage wasn't likely a very healthy marriage. Her wedding band sits "heavily" on her hand and is like a "massive weight." That doesn't exactly hint at a happy marriage.
The final stanza doesn't shift away from Aunt Jennifer or her marriage, but it does shift forward in time. The speaker is looking ahead toward when Aunt Jennifer is dead. The first two lines of this stanza are a rather morbid sequence, but then the final two lines of the poem return to being quite hopeful. Aunt Jennifer might be dead, but the tigers that she made will continue to go on prancing. They will be "proud and unafraid." This is an important piece of the poem because it contrasts with Aunt Jennifer and her horrible marriage. Her marriage and life might have been terrible, but her spirit will live on forever in those optimistic and happy tigers.