I originally learned this was an effort to put their Baltic fleet in ice free ports in case of another flair up with England and France. Lately I've read it may have been a greater show of force to discourage England and France from providing greater support to the Confederacy, and to show support to the United States.
The US interpreted it as a show of support for the United States against the Confederacy. Although the Russians did favor the US over the Confederacy, the real reason for the naval forces was a secret war plan in case of a war against Britain. This was only revealed when the orders were declassified 50 years later.
The Union thought that they had at least the moral support of Russia during the civil war.
Russian reaction to and involvement in the US Civil War is an interesting story.
The Russians actually sent a considerable portion of their navy to the USA in 1863. Russian battle fleets spent almost a year in New York and San Francisco during the Civil War. The Union took this as a sign of Russian support for their cause, and there were a lot of receptions and parties for the Russian officers.
To some extent, the Russians did support the Union. At one point, the Russian admiral in San Francisco issued standing orders to his battleships to engage and destroy any Confederate raiders entering San Francisco Bay. Later, after clarifications from home, these orders were altered to, maintain neutrality unless a Confederate raider actually bombarded San Francisco. Then, on "humanitarian grounds" the Russian battleships were ordered to blow the transgressing hypothetical Confederate raider out of the water.
No Confederate raider actually entered San Francisco Bay.
In actual fact, and contrary to American beliefs, the Russian presence in US harbors was not caused by a desire to show support for the Union. It was part of a Russian battle plan for a feared war against Britain.
The real Russian reasons for their fleet deployment only came to light 50 years after the visit, when the Tsar's secret orders to the fleet were revealed in the Russian archives.
In 1863, there was an insurrection in the part of Poland ruled by Russia. Russia felt it likely, almost inevitable, that Britain and France would go to war in support of Poland (Russia had recently fought the Crimean War against Britain and France from 1853-1856.)
Russian admiral Krabbe proposed a bold naval plan to the Tsar. Instead of letting the inferior Russian navy be uselessly trapped in their ports in the Baltic and Pacific, he proposed sending them to the USA to wait for telegraphic confirmation of war. Then they would burst forth to raid British commerce and bombard British colonies, and contribute to the war effort instead of being blockaded.
These were the secret orders which the Tsar sent to his fleets. So, they sailed to America, and waited for news of declarations of war. The war never came, and after a year, the Russian fleet sailed home again.
The incident led to considerable goodwill between the USA and Russia. It contributed, a few years later, in 1867, when Russia, still worried about war with Britain, thought it might be better to sell Alaska than lose it in war, to a Russian offer to sell Alaska to the USA.
The offer was accepted.