It was intended to try him in public court for treason. They couldn't try him in a closed military court (Like Major Wurz) because Davis was a major political figure and the federals needed to show that the Constitution was the law of the land. It had to be a public trial to justify the war and expose the south's crimes.

However, in order to do that, they not only had to PROVE those crimes, but they had to prove that the confederacy was an illegitimate country and they had authority over them. In order to do that, they had to prove that secession was illegal in 1860 and 1861. This was necessary whether the goal of the war was "preserving the union" or "freeing the slaves". If the confederacy was a foreign country, the federals had no right to do either, no more than the USSR had the right to invade and subjugate the former regions of the Russian empire, or La Raza has in claiming US land.

The question you should be asking is "why didn't Lincoln use his much-vaunted wit, intelligence and compassion to TALK to the three diplomats Davis sent to Washington and convince them that there was no right of secession? Instead, he forbade anyone from talking to them and conspired with the secretary of war and of the navy to invade South Carolina.
Read the Tenth Amendment to the US Constitution, which says that any power and authority not delegated to the federal government is reserved to the states or to the people.

You should also note that Lincoln refused to allow a general surrender of the Confederate government because he thought that would be recognition of their sovereignty. Lincoln would only allow surrender by individual solders and units. Therefore, the war didn't end with Lee's surrender as we have been told for over a century.