This was due to a combination of economic ties, vast oceans, moral reasoning, and geopolitical situations. The entire situation was very complex due to the number of actors and some strange situations and ideas emerged. For example, would it surprise you to find out that one of the earliest ideas was for the northern states to provoke a war against Britain that would force the south to rejoin the union? This idea didn't grab much traction for obvious reasons.

The Southern states had the most to gain with foreign help. Just like in the American revolution, the initial goal was recognition, which would help considerably with financial pressures, but also with military intervention. The south believed in the foreign policy concept of "King Cotton", namely that the United Kingdom and France would recognize and aid the Confederacy due to the dependency of their economy on imported American cotton. Many southern states self-embargoed their own exports of cotton at the beginning of the war to prove this point. This in turn failed as the UK took the economic hit and switched to cotton from Egypt and Argentina, while the southern states lost valuable capital that the goods would have delivered.

The next issue was the vast ocean. The UK once again was the dominate player in both military and trade matters, but the Northern States maintained a significant navy, and while American trade would atrophy badly during the war and for generations afterwords, it affected the south far more then the north. The Northern naval blockade badly restricted confederate trade; this removed commerce as justification for war. In the end the United Kingdom benefited as both north and south trade was severely restricted, as insurance providers were no longer willing to cover American ships (north or south) and as a critical trade dependency (cotton) was replaced by more friendly providers.

The northern blockade also made it harder for the confederates to interact with European governments. This backfired almost catastrophically with the Trent affair, where a American ship intercepted a British packet ship, and removed two confederate agents from the ship. The UK took that badly, and went onto a war footing with plans to seize New York. The Northern States deftly handled the reaction, in particular disavowing the seizure, and agreeing to several international agreements. This ended up being a setback to Confederate plans over the long terms, as Europe generally felt that the best way to avoid conflict was to embrace neutrality and work with the Northern States.

One of the treaties that was signed was a long standing British goal - namely the United States finally rejected the Slave Trade in the Lyons-Stward Treaty. This was a fundamental problem that also kept the UK from recognizing the United States. The UK had been the strongest voice against slavery since 1833. Once the emancipation proclamation was issued, for them to recognize or aid the confederacy would be inconsistent with their moral stance on slavery. Even the regions hardest hit by the "cotton famine" such as Lancashire and Manchester chose to support the union despite the economic hit.

The geopolitical constraints dictated the response as well. UK recognition of the South would also cause massive issues with both Scotland and Ireland, to say nothing of the rest of the Empire.

Of the European nations, the only other nation that could play a potentially decisive role in the Civil War was France. France was considerably weaker outside Europe then the United Kingdom, and it was inconceivable that France would go to war and assist the confederacy without the aid of the United Kingdom. What's worse is that Cotton was king here, but not in the confederacy's favor. The Union Blockade wiped out Southern Cotton, and the economy of France took a hit from "The Cotton Famine". The Southern states offered to sell raw cotton to France, but the blockade and the capture of New Orleans made this a promise the southern states could not fulfill. In addition if Napoleon III of France did engage in the United States, it would open the door for military adventures from Prussia.

So instead of aiding the confederacy openly, France chose to invade Mexico and lend money to the confederate states without formally recognizing the rebels. In this they hoped to take advantage of America's inability to militarily oppose the invasion, without giving the United States a reason for war. Once the civil war ended, France's would abandon the war in favor of normalizing relationships with the United States.