You have to realize that Europe was ruled by interemarried families, most of which traced their origins to the Merovingian Kings. The Hapsburgs, the Bourbons, etc... all belonged to that ROYAL line. So they stuck together to help maintain their power. That doesn't mean they wouldn't fight amongst themselves, but they definitely all communicated with each other and did their best to maintain the aristocracy.
When Louis was arrested, most of the European Monarchs believed that he probably did a poor job and weren't extremely sympathetic to him. However, he was still a King and of Royal blood and in protecting him to an extent, they were protecting themselves. So they made it very clear that they would not interfere in France's internal politics. However, should anything happen to the King and Queen, that would be a different story.
They assumed that the French Revolution was a passing phase and eventually a King would rule France when the internal problems were resolved. However, when he was killed, that was a threat to all of their thrones and they greatly feared the expansion of the Revolution to their Kingdoms. This was a very specific threat of the Revolution and they paid attention to it after the death of Louis and Marie. So France was attacked and that was the state that France was in when Napoleon Bonaparte came on the scene.
For nearly 20 years France was entangled in nearly non-stop war with the various European powers as a result of the removal of the Monarchy. Had France allowed King Louis to live essentially under house arrest somewhere and was treated reasonably well they would not have intervened at that point. Eventually, something would have had to be done, but killing Louis brought the issue to the forefront immediately.
