The sentence is drawing a parallel between the two parties' reasons for going to war. Each of them is associated with a verb: one with "survive," the other with "perish." They are the two antonyms.
Antonyms are words which have opposite meanings. Survive means to contine living, and perish means to stop living.
What Lincoln meant by using this antonym-based parallelism is that the Confederacy fought to end the exisiting nation (meaning, to secede from the North and create a new nation), while the Union fought to keep the nation alive by preventing it from splitting.
