More can always be said but I've written on the topic of Mexico in that period before which I'll repost below:

The summer of 1861, with the 'War of Reform' barely in the past, was a fairly chaotic one for Mexico, with the threat of renewed internal conflict high. The concluded conflict had drained the treasury, and placed the government severely in debt to foreign creditors. This was further compounded by the significant damage to the property, especially British, for which the new government of Benito Juárez was being held accountable to provide full repayment. The government was broke though, so Juárez called for a pause on repayment for the next two years, which certainly pissed off Europe, especially France, Spain, and the British.

While Mexican debtors included the United States, under normal circumstances the European powers would have been limited in the degree to which they could have pressed for repayment, knowing full well that too strong a show of force would result in angering the United States, if not military intercession. But the United States was busy with its own issues at that time, unable to rattle its sabres in defense of the Monroe Doctrine, and as such, Napoleon III saw his chance, proposing that the three powers join forces for a demonstration of power against Mexico to force repayment, which would then allow for the invasion of the country to place Ferdinand Maximilian in power as Mexican Emperor, something of a dream for Napoleon, who envisioned a French-influenced puppet empire on the American continent.

Britain and Spain mostly just wanted their money. As such, they committed a naval contingent, and the British government sought assurances from the French that this would be the extant of the exercise. Show up, remind Mexico that they needed to pay their debts, and go on their merry way. Napoleon III assured them he would abandon part two of the plan and then promptly returned to planning the invasion and just didn't let the British know. The Tripartite fleet arrived at Veracruz in December, 1861, and land troops put ashore the next month, and issued their demands that debts obligations be met.

Not acting exactly in good faith, after all they had shown up with a navy behind them, the British and Spanish ministers, Charles Wyke and Juan Prim, were satisfied with the terms negotiated at swordpoint, and by April had ordered the respective forces to withdraw. The French at this point made their real intentions known, not only refusing to leave, but marching inland to begin their conquest of the country. The Mexicans put up a fierce resistance, flummoxing French advances at several points (including the victory on "Cinco de Mayo"), resulting in an entire army corps being sent to reinforce the invaders, and it wouldn't be another year until Mexico City fell on 10 June, 1863. It would be the better part of a year before Maximillian would take the offered crown, in large part recognizing the displeasure by the British over what had happened, finally convinced by an obviously forged petition claiming 6.5m votes in favor, allowing him to become a French puppet who ruled exactly as much Mexican land as there were French troops to occupy.

As for the United States in all this, to say they were upset would be an understatement, but they found themselves in an unfortunate position of relative powerlessness. The Tripartite actions had been a thumbing of the nose at the United States as it was, while the ensuing French invasion was a full slap in the face. To head off the possibility of European action, it has even been proposed that the United States take on Mexico's debts by offering their own loan of $12m to allow repayment to the European debtors, but it was voted down by the Senate, although there was also no awareness of the extent to which the French were aiming to go at that point.

There were also concerns, not unfounded, that Napoleon III would push for recognition of the Confederacy as a legitimate government if it meant destabilizing the United States to serve his own ambitions. In the end never materialized, but the United States government realized that it couldn't push back on the French invasion harsh enough to risk recognition of the Confederacy, let alone war, as it could spell their own doom. Even though Napoleon III continued to deny his full intentions right up until Maximillian's crowning in 1864, the Americans had little credulity for such claims, but also little they could do. The American press though could at least be more vocal, headlines crying out against the invasion, and promising that once their own domestic issues were handled, the American forces would make quick work of the French interlopers. And while unable to do much action, the Lincoln administration did nevertheless provide financial backing to the resistance forces under Juárez.

The slight irony of course was for the Confederacy, hoping for that possibility, it meant biting their tongue about the French actions and its potential for further encroachment in the continent, although it didn't stop a few voices being raised that it would be best to put aside differences, combine forces, and kick them out of Mexico (the most notable being from Alexander Stephens to Lincoln in early 1865. Too little, too late by then). Perhaps more amusingly, later on in the war, there was some vain hope that Napoleon would recognize the Confederacy because he could call on them to assist in the pacification of Mexico, as things continued to not quite work out.

Mexico was never truly pacified, and in 1865, the United States had managed to sort out is little domestic tiff, and was able to finally not only bring about the full diplomatic pressure it has been eschewing the past few years, but do so with a large, battle-tested army giving clear weight to the matter, including 25,000 troops sent to the Mexican border at the beginning of the summer. In the end though that was only one of several factors in the French withdrawal. Realizing his dream of a French controlled puppet state in Mexico was slipping from his fingers, the concerning rise of Prussian power nearby was perhaps more pressing that that of the United States abroad, and Napoleon III was coming to see that Mexico would likely be a bottomless pit insofar as its long term requirements for French funding to keep propped up, for what he had thought would be a project of a few mere months several years prior.

As such Napoleon III cut off his support for Maximillian, in a ludicrous declaration that "France had accomplished its objective for civilization" and that Mexico had been transformed into "a regular power which was ready to fulfill its commitments", French troops were recalled in January, 1866, leaving only the Mexican monarchists who had chosen to back the foreign horse. The end for Maximillian was only a matter of time at that point, managing to hold on until mid-May of 1867, after which he was captured, tried, and executed a month later.

Sources

Cunningham, Michele. Mexico and the Foreign Policy of Napoleon III. Palgrave Macmillan, 2001.

Fehrenbach, T. R.. Fire & Blood: A History of Mexico. Da Capo Press, 1979.

Jones, Howard. Blue and Gray Diplomacy: A History of Union and Confederate Foreign Relations. The University of North Carolina Press, 2010.

Sainlaude, Stève. France and the American Civil War: A Diplomatic History. University of North Carolina Press, Apr 2019.