Why don't the stars in a star cluster attract each other gravitationally, forming one big star? What causes a cluster to disperse the stars in it?
"Dissipate" and "disperse" are the wrong way to approach this context. It implies the cluster undergoes a more compact state, followed by expansion - which is not the case, or not always.
First off, big blobs of gas don't form a single giant star for the same reason why the whole Arctic Ocean doesn't form a single giant iceberg - there's too much local motion for that. Instead, both gas in a proto-cluster, and water in the ocean, coalesce into smaller chunks - the individual stars, or the ice floes.
Following that, the evolution of a star cluster is very diverse. Sometimes they do undergo a sort of "compression", where there is a global motion of the stars towards the center. Other times they inflate and end up occupying a larger volume. Other clusters pulsate. Yet others are seemingly chaotic (although there's nothing truly random about the motion of each component star).
The stars in a cluster don't simply fall into each other for the same reason why the planets don't fall into the Sun: they are orbiting their common center of mass. In a cluster, each star follows a path described, in broad terms, as an orbit around the cluster's center; however, the nearest neighbors act as perturbations, and so most stars don't actually circle the center on perfect conic-section curves.
The dynamics are pretty complex and there's no one pattern to rule them all.
I suggest you download Universe Sandbox and run one of their cluster simulations.
The short-short answer is: angular momentum and light pressure.
When a cluster is forming from interstellar gas
Once the stars are formed, their orbital interaction will preserve both angular momentum and energy--if their orbits bring them closer together they must speed up (this is just a more complicated version of Kepler's Laws), and unless they ram head on into one another they will take that speed and head on out again.
And the physical stars are miniscule by comparison to the volume the cluster occupies. They will rarely intersect even on cosmological time scales.