While what a poet is trying to tell the reader will usually be a matter of dispute, B. C. Southam shines some light on these lines in A Student's Guide to the Selected Poems of T.S. Eliot, pp. 217-8:
ll. 95-8: a parody, combining a line from the children's song 'Here we go round the mulberry bush' - 'This is the way we clap our hands' -
with a distortion of the phrase 'world without end' from the prayer
'Glory be to the Father, and to the Son, and to the Holy Ghost, as it
was in the beginning, is now, and ever shall be, world without end.
l. 98: Not with a bang: cf. 'it all ends, not with a bang, not with some casual incident, but in sustained reflection...', in George
Santayana's account of the Divine Comedy, in Three Philosophical
Poets (1910). Santayana (1863-1952) was lecturing on Dante at Harvard
during Eliot's student days.
whimper: Eliot may have had in mind two lines from Danny Deever (1892) by Rudyard Kipling, a poem he knew by heart at the age of ten. Deever, a British soldier, is executed in front of his regiment for killing another comrade:
What's that that whimpers over'ead?, said Files-on-Parade,
It's Danny's soul that's passin' now,' the Colour-Sargeant said.
Eliot referred to this 'remarkable' poem in his Introduction to A Choice of Kipling's Verse (1941) quoting these lines and commenting that Kipling's choice of the word 'whimpers' is 'exactly right'.
The 'whimper' may also combine an allusive reference to Dante, suggesting the cry of a baby, the new-born leaving one world to enter another, just as at the end of the Purgatorio Dante is leaving the sinful, fallen world in the presence of Beatrice in the Earthly Paradise. In xxx, 43ff. and xxi, 64ff, he stands before her, shamed and conscience-stricken, repentant and silent, like a child before a stern mother.