Determining why one species (or group of species survived) while others did not can be difficult to answer, but I can offer one possible explanation for the genus Homo: competition. Competition between similar species is one possibility. Consider for example gorillas or chimps. The two species of gorillas occupy different habitats so they rarely compete with each other for resources (food, water, shelter, etc.). The two species of chimpanzees have non-overlapping ranges (or minimally so) so again they do not compete for resources.
Chimps and gorillas do overlap with each other for parts of their range and even potentially use the same resources, although no competition for resources has been observed (see this primate factsheet and McNeilage 2001). The same applies for the orangutans, which lives on two different southeast Asian islands.
The genus Homo is an entirely different story. As you may know, Homo first evolved in east central Africa. The first species to leave Africa and disperse across Europe and Asia was Homo erectus. This migration out of Africa happened as early as aout 1.8 million years ago. Homo sapiens had not yet evolved. However, while H. erectus was establishing itself across Europe and Asia, populations still in Africa continued to evolve.
Modern Homo sapiens evolved in central Africa perhaps about 250-200,000 years ago and dispersed out of Africa about 100,000 years ago. Homo sapiens also dispersed across Europe and Asia. Both H. erectus and H. sapiens co-occurred across parts of their range. Some have argued (Stringer and Andrews, among others, I believe but I'll have to be at work before I can verify ths reference) that H. sapiens outcompeted H. erectus, leading to its eventual extinction by competition. They occurred together and used the same resources, so H. sapiens could have caused the extinction of H. erectus. This has been called the replacement hypothesis. The later evolving species (H. sapiens) replaced the earlier evolving H. erectus.
However, Templeton (2002) has argued that the two species interbred so the species became integrated rather replaced (if I am interpreting him correctly). While I don't dispute the possibility of interbreeding, I am skeptical of Templeton's complete dismissal of replacement.
Complicating the picture is H. neanderthalensis. Genetic evidence published by Green et al. (2010) suggests that Neanderthals diverged from H. sapiens (in central Africa) between 800,000-500,000 years ago and also dispersed out of Africa. They also provided evidence of interbreeding between H. sapiens and H. neanderthalensis in Europe. Modern H. sapiens may also have outcompeted H. neanderthals.
All of this was occurring during the Pleistocene ice age. Europe was experiencing repeated bouts of extensive glaciation. Thus, it is also possible that climate change caused or contributed to the extinction of H. erectus, H. neanderthalensis, or both.
In summary, the processes that led to the extinction of other species of Homo are complex and likey interactive. Competetion among species is, in my view, well within the realm of possibility but would be extremely difficult (if not impossible) to test. Climate change is also a very likely contributing factor. There are several other hypotheses but I have tried to hit the major highlights. The debate is certainly not settled.
McNeilage A. 2001. Diet and habitat use of two mountain gorilla groups in contrasting habitats in the Virungas. In: Robbins MM, Sicotte P, Stewart KJ, editors. Mountain gorillas: three decades of research at Karisoke. Cambridge (England): Cambridge University Press. p 265
