This passage is clearly meant as Lincoln's expression of how he thinks that the Union should move toward the ending of the Civil War and the Reconstruction of the South.
When Lincoln delivered this speech, the Civil War was about a month from begin over. It was clear that the Union would win. The North, then, had to decide how to treat the South after the war. Lincoln was giving his basic view on how to do this.
He was saying that the North needed to remain "firm." This meant that they could not give up on the purpose of the war, which was to preserve the Union and, by this point, to end slavery. At the same time, however, he did not want to be harsh in dealing with the South. This is why he talks about acting with a charity and without malice.
Lincoln is proposing, then, a fairly lenient attitude towards the South so long as it is compatible with achieving the goals of the war.