What does lincoln mean when he says "neither anticipated that the cause of the conflict might cease with, or even before, the conflict itself should cease" (lines 30-32)

The main cause of the war was about slavery. There are those who are for slavery and those who are against it. Their belief and interest resulted to the waging of the civil war, Union vs Confederates.

However, slavery was abolished before the war came to its end.

The 13th amendment which abolished slavery was passed on January 31, 1865.
The Civil war ended on May 9, 1865. More than 3 months after the passing of the 13th amendment.

To get what you want you have to work for it.