Why do most laser beams have a Gaussian intensity profile?

Recall that a laser is an oscillator composed by an amplifier, the active medium, and a resonator, the optical cavity. The light emitted by the laser is a combination of the cavity modes, which combination depends on the cavity properties and the gain profile of the laser.

The intensity profile is thus determined by this combination of modes. If you consider for example an optical cavity composed of two spherical mirrors, it turns out that its modes are Gaussian beams: the fundamental mode has an intensity profile which is a simple Gaussian curve, while higher modes have more complicated profiles (see e.g. this document).

Hence, a so-called single-mode laser working on the fundamental mode of a cavity with two spherical mirrors, will have an intensity profile quite close to that of a Gaussian.

In general, not all lasers are single mode and not all lasers are equipped with those kind of cavities (e.g., solid-state lasers). Therefore, not all laser beams are perfectly Gaussian. However, designers always try to choose cavity shapes which minimize the field at the edges of the cavity to reduce diffraction losses.

While most lasers generate Gaussian beams, for reasons well outlined by Massimo Ortolano in his answer, this is not the only possibility. Other two kinds of laser profiles that have applications in optical laboratories are for example Hermite-Gaussian and Laguerre-Gaussian beams.

The latter are in particular very interesting, as laser light with a Laguerre-Gaussian amplitude distribution happens to have a well-defined orbital angular momentum, as first observed by L. Allen in 1992. Quoting from Allen's paper: The transverse amplitude distribution of laser light is usually described in terms of a product of Hermite polynomials $H_n(x)H_m(y)$ and associated with TEM$_{nmq}$ modes. Laguerre polynomial distributions of amplitude, TEM$_{plq}$ modes, are also possible but occur less often in actual lasers.

Even when the laser itself can only generate a fundamental Gaussian profile, the spatial profile of the light is (relatively) easily modified, for example with SLMs, and made into any profile one wants.