Why does a singularity need to exist at the center of a black hole?

The singularity is not a physically established fact but a prediction of a classical field theory (that's what general relativity is). Such predictions are usually seen as a sign that a theory breaks down in a certain regime or on a certain scale and that it has to be replaced with a better theory.

Unfortunately for us, no measurement has been made to date that contradicts general relativity, so we simply do not even have a slight hint of evidence of what to replace it with. We can speculate about what happens near the singularity predicted by general relativity and have done so plenty, but there is, at this time, simply nothing available that could tell us which of these speculative solutions is the correct one, or if we need something completely different that nobody has guessed, yet.

As for beliefs... that's not how science works, so there is no need to garner any. Science starts with empirical evidence and then compares it with hypothetical explanations. The explanation that fits the evidence best is eventually elevated to the rank of a theory. As of now we simply do not have a good theory of what is "inside" a black hole. "We don't know" is therefor the correct and perfectly adequate scientific answer. We think we understand what general relativity predicts to be in there... and that's called a singularity (and it's not even a point like object if I understand some of these predictions correctly). I doubt that, figuratively speaking, more than a handful of physicists think that's the entire story, though.

It might be worth looking at this in the context of a simpler theory of gravity - Newtons; there it is said that the gravitational attraction between two particles is inversely proportional to the square of their radial separation.

Thus the closer they are, the stronger the attraction; so what happens when they get infinitely close? We have a singularity ...

Singularities tend to signify that there is a breakdown of some kind in our theories of reality.

Here it is easily fixed: particles cannot get infinitely close to each other - they are charged so at as certain point repulsion kicks in; another possibility is that extensionless particles are not possible - so strings; which resolves some of the infinities in a perturbation expansion ie Feynman diagrams.

The point then, is that somehow one has to resolve the singularity in a black hole; this will most probably be part of a full theory of QG - an ongoing project.

Both String theory and Quantum Loop Gravity, apparently have a 'microscopic' picture of black holes; this conjecturally explains black hole entropy - macroscopic datum - found by semi-classical means, by a detailed picture in the small; this then, suggestively resolves the singularity.

But given the traditional conception of physics as an empirical science; the conjectural status of both these theories of QG - it can hardly be said to be entirely conclusive; but it is certainly suggestive.

The other answers have done a good job of covering the fact that we don't know if singularities actually physically exist, they might represent a breakdown of our theory, etc.

Here's another point which I think is worth making. Within the theoretical and mathematical framework of general relativity, singularities are guaranteed by the singularity theorems of Hawking and Penrose. Roughly, these theorems require the following things to hold:

-The spacetime you're working with satisfies certain energy conditions (we can use very weak ones to prove the singularity theorems, they're pretty physical and not very stringent).

-The spacetime satisfies certain global causality conditions (essentially guaranteeing no closed timelike curves allow time travel to the past).

-There exists a surface in the spacetime from which we could emit light and it could not escape to infinity (a formal way of testing that our spacetime contains a black hole).

If these things hold, the theorems guarantee that a singularity exists! So within the context of general relativity, singularities are inevitable. This was not obvious at first, and people worried that the high symmetry of our exact black hole models, and that maybe realistic black hole collapses might not have singularities. The singularity theorems well us that this is not an out. Very symmetric models do not explain away singularities, the conditions for the theorems to hold are very flexible.

Now, this guarantee requires that general relativity be a final theory which won't fail in the sort of regimes where singularities form. We think general relativity will probably have to be modified into a full theory of quantum gravity to make real predictions inside black holes, but general relativity taken literally requires singularities. It gives us reason to suspect that even if singularities don't survive in the full theory, they almost do in the sense that there will be extremely dense regions in the interiors of black holes.