Why doesn't the nuclear fusion in a star make it explode?

The fusion that occurs in the core of the Sun occurs in nothing like the conditions you might be thinking of in a bomb, or a fusion reactor. In particular, it occurs at much lower temperatures and at a much lower rate. A cubic metre of material in the solar core is only releasing around 250 W of power by fusion.

The fusion rate is set by the temperature (and to a lesser extent, density) of the core. This in turn is set by the need for a pressure gradient to balance the weight of material pressing down on it from above. At 15 million kelvin (the core temperature, which is much lower than the temperatures in nuclear bombs or fusion reactors), the average proton has a lifetime of several billion years before being converted (with three others) into a helium nucleus. There are two reasons this is slow. First, you have to get protons, which repel each other electromagnetically, close enough together to feel the strong nuclear force. This is why high temperatures are needed. Second, because the diproton is unstable, one of the protons needs to change into a neutron via a weak force interaction, whilst it is in the unstable diproton state, to form a deuterium nucleus. This is just inherently unlikely and means the overall reaction chain to helium is very slow.

The reason there is no bomb-like explosion is because there is no problem in shifting 250 W per cubic metre away from the core, in the same way that a compost heap, which generates about the same power density, does not spontaneously explode. In the case of a star any additional heat goes into more radiation that diffuses away and in work done in expanding the star. As a result, the temperature of the core is stable. Ultimately, any additional energy emerges as sunlight at the solar photosphere.

If for some reason, the opacity to radiation in the core increased, then the temperature would rise and more energy would be generated by fusion. This is exactly what happens in the core as more hydrogen is turned into helium; the core temperature and luminosity do rise, but slowly, on timescales of billions of years.

If fusion were to proceed faster, the core would get hotter, it would expand and become less dense, and with less density, fusion would slow down.

The main sequence in stars like the Sun does proceed much more slowly than other stages. This is because the p-p chain reaction starts with the fusion of two protons to form a diproton, or helium-2. The diproton is unstable, and usually immediately decays back into two protons, but Bethe realised that on rare occasions it decays by a weak reaction, releasing a neutrino and a positron to form a deuterium nucleus, hydrogen- 2. Because this second process is so rare, it limits the rate of stellar fusion so that stars spend the largest portion of their lives on the main sequence

Fusion in stars requires enormous pressures and temperatures.

Any body, including stars, are subject to their own gravitational field. At any point inside a spherically symmetrical body (which most stars approximate well) the gravitational force will be due to all the mass "below" that point - between that point and the center. That gravitational force obviously points inward.

However all the mass outside that radius is also being pulled inward and exerts pressure on the material below it. This adds to the gravitational force of the material inside.

So enormous pressures exist at the core. As the pressure increases the conditions for fusion become more and more likely. When fusion happens the core region that can allow fusion is kept contained by the pressure from the material outside that core, which cannot fuse. Note that fusion doesn't happen everywhere in the star, just at that core region which has reached high enough pressures.

The energy being generated by fusion keeps everything hot (simplistically) and hot things like to expand and produce an outward pressure. It's the outward pressure from fusing core's thermal energy (which is passed by radiation and convection throughout the star and eventually outside the star as light) that prevents the gravitational collapse of the core due to the force of everything pressing "down" on it.

So it's the gravitational force the body exerts on itself that prevents it "exploding" because it causes fusion which generates heat that pushes against the collapse.

Why is the nuclear fusion happening slowly?

Slow is a relative term, but the rate of fusion is decided by the pressure and temperature inside the star. Oddly enough smaller stars tend to live the longest. This, very simplistically, is because the pressures at the core are relatively low and the amount of fusion that can be maintained by it and the size of the fusing core are correspondingly small. Larger stars have more pressure and larger cores and can burn relatively quickly. The detailed reasons behind the lifetime of stars are somewhat more complex. If you want to read more about this I'd suggest reading e.g. Wikipedia's pages on red dwarf stars and Stellar Nucleosynthesis.