In something such as a sound wave it is easy to understand. The air is made up of molecules and atoms. These can impact each other at many different angles and over time transfer the energy/momentum of the string's motion outwards in all directions. However, this does not work for something like an electromagnetic wave. In that case you need to look at the huygens-fresnel principle, linked above, which applies to all kinds of waves.
Basically the principle boils down to the fact that a disturbance in a medium or field would rather propagate outwards in all directions if it capable of doing so. If we had an infinitely small. point-like oscillator, this would indeed be the case, and you'd get a perfectly spherical wavefront. But with real, extended oscillators, such as a vibrating string, the disturbances from different parts of the medium/field interfere with each other and prevent themselves from spreading out. On the boundaries of the oscillator (and resulting disturbance), the disturbance can indeed spread outwards since it lacks something to interfere with.
The Huygens-Fresnel principle describes all of this by taking a wavefront (the disturbance) and modeling it as a sum of an infinite amount of 'wavelets', which are waves generated by a point oscillator. The interference of all these wavelets gives you the resulting wavefront.
Your original question, which I take to be "why does it spread out in the first place", is mostly just an observed fact. When the oscillator is very small compared to the wavelength of the oscillation, the resulting wave spreads out much better than when the oscillator is very large compared to the wavelength of the oscillation. In other words, the smaller the oscillator is compared to the wavelength of the oscillation, the more closely the oscillator approximates a point-like source. This is one reason why directional antennas have to be a certain minimum size in order to work correctly.