Meaning
"Nor even the tenderest heart, and next our own, Knows half the reasons why we smile and sigh."
(Compare 1Ki 8:38.) Something there is in every sorrow which no one else can share. "And a stranger doth not intermeddle with its joy," not necessarily in an interfering or any offensive way, but simply does not share or take any interest in the other's joy.
For "intermeddleth with" (Pr 18:1 the King James Version), the Revised Version (British and American) gives "rageth against" (margin "quarrelleth with").
M. O. Evans