There is a lot of depth to this issue. In short, the international community thought Rwanda was insignificant and hopeless so they didn't feel like it was worth their time. This decision was later regretted by much of the community. The deeper explanation is below.
Since the decolonization of Africa after the Second World War, there has been an great number of wars between ethnic groups in Africa. This has origins at the Berlin Conference of 1884 where the superpowers of Europe essentially drew lines on a map of Africa and claimed bits and pieces of it for themselves. This created the modern day African countries, mixing ethnic groups and races that historically hated one another. This, as one can assume, caused many problems once the colonial powers were no longer there to enforce peace. Especially since the colonial powers had granted certain groups elite power, generally those who were seen as being more "civilised", i.e. most like White Europeans [ie the Tutsi] and these minority tended to rule quite cruelly over the majority.
This was the case with the Belgian Empire in Rwanda. The Tutsi, a light skinned group, was granted elite status over the Hutu and the lesser known Twa ethnic groups. It remained this way until a Hutu rebellion in the 1950s [After the Belgians had left] in which the Hutu rose up and slaughtered 10,000 Tutsi and took power. Many more Tutsi fled to Uganda to escape annihilation. Habyarimana came into power in the 1970s and he had a very friendly relationship with French president Francois Mitterrand. Mitterrand made very many of his diplomatic decisions in Africa based on what is colloquially referred to as "Fashoda Syndrome", a mindset that there is an Anglo-Saxon conspiracy to eradicate French culture from Africa (born of a conflict in Fashoda, Sudan in the 1890s). This friendly relationship, mainly driven by Habyarimana's francophone government that supported a quasi-democracy in Rwanda, continued well into the 1990s and even after Habyarimana's death. The relationship manifested itself into arms, military training, and even troops both before and after the RPF invasion that began the Rwandan civil war and well into the genocide. After all, it was the French that originally trained the Interahamwe. Now you see why France, one of the permanent seat holders on the UN security council, never voted to upgrade to a Chapter Seven Peacekeeping mission, one that would allow the peacekeepers to use weapons, on Romeo Dallaire's request.
Now to move on to Belgium. The Belgian failure to respond is very easily understood. They were a part of the peacekeeping mission and even willing to help out until their 10 soldiers that were peacefully guarding the Prime Minister were tortured, maimed, and killed. Thus Belgium pulled out of the UN mission because the general Belgian public was unwilling to lose any more men. At that point they had no idea that the country was going to descend into an inferno of murder and death and they were more concerned with preserving public opinion.
Very few people take the time to think about the mindset of the United States at the time. In order to fully analyze the motivation behind their indifference, one must look at the events that were occurring. The most famous being Somalia, where a mission that was to be pretty much a "hole in one" ended in casualties that should never have occurred. This naturally was a bit of a blow to the United States, who had invested countless amounts of troops and even more money to peacekeeping missions in Africa, some of which had gone very terribly wrong. Therefore when the Rwanda conflict occurred less than a year later, the United States was still stuck with "We can't have another Somalia" because they realized how detrimental that could be to the US Government. The way they saw it was that no matter what they did, it would not change the situation in Rwanda and it would only cost American lives without accomplishing anything. Yes, the US voted against sending 2,500 troops like Dallaire requested and, being a permanent seat holder on the Security Council, it made a big difference. But to blame all of this on the US is not justified.
The British turned a blind eye for nearly the same reason.
The fact of the matter is this, Rwanda has no natural resources and very little to offer to anyone but the French who needed another francophone country to battle the "invasion of the Anglo-Saxons". And even the French did not care. Francois Mitterrand once said that "in a country like that a genocide does not matter" which, for the most part, fully encompasses how the international world felt about Rwanda at the time.