What allows massive new born suns to move away from each other, as they have been observed. I would think that their massive gravity would prevent this and cause them to slam into each other.
Newborn stars often do remain close together as gravitationally bound, binary systems. They can't crash directly into each other because of conservation of angular momentum in the binary system. As in all problems involving orbits, it is instructive to consider the effective potential experienced by one of the bodies. The familiar gravitational potential term is present, but there is also a repulsive centrifugal term.
Generally speaking, stars form in turbulent clouds of gas. Sometimes the collapse of the cloud will yield only a single star. More often it will fragment into multiple young stars, some of which may form binaries and some of which may drift apart in the combined gravitational field of all the stars in the galaxy.
There are three ways I'll describe in which stars can be dispersed in a cluster: multiple-body interactions, stellar winds and supernovae.
First, multiple-body interactions. Suppose a single star approaches a binary system. i.e. a pair of stars orbiting each other. There's a big range of outcomes of the interaction but many involve the orbits of the first two stars becoming tighter, especially if they are more massive than the interloper. The energy that is lost in the contraction of the orbit is imparted on the third star, which gains kinetic energy. That is, it flies off. This is expected to happen at the cores of dense star clusters.
Second, stars give off material winds, driven by some of the energy released in nuclear reactions in the core. This has two effects. First, it drives gas out of the cluster, so that all the objects are less strongly bound because there is simply less gas. Second, it decreases the mass of the star, so if it's in a binary, the binary orbit widens.
Third, suppose you have a binary system where the stars are far apart enough that they don't interact during their lives. The larger star evolves first and might eventually explode in a supernova. In this case, there's at most a small remnant that's left to hold on to the other star, so it can if effect fly off at nearly it's orbital velocity. Think of swinging a ball around on a string and then letting go. These events are known as supernova kicks.
I've only got time to ink the answer, sans links, so I encourage anyone passing by to add some.