Why are stars so far from each other? Shouldn't gravity pull them closer over time? And if the effects of gravity are negligible is there an explanation why stars have to be so distant from one another?
The closest star is Alpha Centauri (I think) and it is 4.4 light years away.
The initial star formation regions were regions that have a high enough mass density to form a star. The density of the early universe was not constant at different locations. Some regions had high enough density to form a star, and some didn't.
When a star forms it draws in matter from a large distance away. This forms an accretion disk and leaves a temporary relatively empty space around the star at a large distance away. There is no way a star can form in this low mass region. Once you get further away from this region the matter density may return to a level where a star can form. But this region is far from a star.
Yes, gravity from a particular star is pulling at other stars but there are many stars pulling on the particular star of interest. So the net force on the star may be very small.
As jmh has answered, stars naturally form at large distances from each other. To add to the answer, what is the reason for this particular distance scale?
If we imagine a very large, homogeneous cloud of gas it will be unstable to gravitational collapse over the Jeans length scale $\lambda_J=c_s/\sqrt{G\rho}$ where $c_s$ is the speed of sound in the gas, $G$ the gravitational constant, and $\rho$ the density. For typical values this about a light-year, giving a sense of the distance scale.
But why are those values what they are? $G$ is a fundamental constant and since it is small we get long distances. The speed of sound depends on temperature and molecular mass; since molecules are very light it is high. Why are molecules light? This is because another fundamental constant, the ratio between proton and electron mass, is large. The density of gas in the universe is low, since the universe has a density close to the critical density (had it not been that, it would either have recollapsed or expanded so fast there would not have been many stars).
A universe where stars form much closer to each other than in ours needs to have strong gravity (making stars burn much hotter and be short lived, beside lots of gravitational interactions disrupting planetary orbits), have heavy molecules (making chemistry weird), or have a high density (likely collapsing rapidly). So it is likely that there would not be any life and observers there.
The question also asks why gravity does not pull them closer to each other. Note that if the cloud turns into stars with typical mass $M\approx \rho \lambda_J^3$ separated by distance $r\approx \lambda_J$ then the acceleration between them will be $a=GM/r^2=G\rho \lambda_J^3/\lambda_J^2=G\rho \lambda_J =\sqrt{G\rho}c_s$. So the accelerations will be very small, again for the same reasons as discussed above.
(Further, due to the ratio between the electromagnetic force strength and the gravitational force strength, objects like stars have equilibrium sizes that are small compared to $\lambda_J$, so they rarely collide with each other on this timescale. They just miss, and fly past. )
All these stars orbit the galaxy center of mass. The galaxy outweights any possible star by many orders of magnitude and dominates the orbital motion, unless some stars come pretty close together.
The orbits of the stars are more or less stable in the same sense that orbits of planets are stable in the solar system. They are not absolutely stable, but stable enough on a tens of bilions of years timescale.
Even if two stars come closer, they rarely collide. There is too much of space and too few stars. In most cases they just miss each other in an open hyperbolic orbits.
It is estimated that if the Milky way galaxy collides with the Andromeda galaxy, there will be less than 10 actual star collisions out of bilions and bilions of stars in both galaxies.
bonus: a pretty graph of nearest stars approaching and passing by the Sun, from Wikipedia: https://upload.wikimedia.org/wikipedia/commons/e/e9/Near-stars-past-future-en.svg
