There are two principle motivations:

First, the beta distribution is conjugate prior to the Bernoulli distribution. That means that if you have an unknown probability like the bias of a coin that you are estimating by repeated coin flips, then the likelihood induced on the unknown bias by a sequence of coin flips is beta-distributed.

Second, a consequence of the beta distribution being an exponential family is that it is the maximum entropy distribution for a set of sufficient statistics. In the beta distribution's case these statistics are $\log(x)$ and $\log(1-x)$ for $x$ in $[0,1]$. That means that if you only keep the average measurement of these sufficient statistics for a set of samples $x_1, \dots, x_n$, the minimum assumption you can make about the distribution fo the samples is beta-distributed.

The beta distribution is not special for generally modeling things over [0,1] since many distributions can be truncated to that support and are more applicable in many cases.

There are two principal motivations:

First, the beta distribution is conjugate prior to the Bernoulli distribution. That means that if you have an unknown probability like the bias of a coin that you are estimating by repeated coin flips, then the likelihood induced on the unknown bias by a sequence of coin flips is beta-distributed.

Second, a consequence of the beta distribution being an exponential family is that it is the maximum entropy distribution for a set of sufficient statistics. In the beta distribution's case these statistics are $\log(x)$ and $\log(1-x)$ for $x$ in $[0,1]$. That means that if you only keep the average measurement of these sufficient statistics for a set of samples $x_1, \dots, x_n$, the minimum assumption you can make about the distribution of the samples is that it is beta-distributed.

The beta distribution is not special for generally modeling things over [0,1] since many distributions can be truncated to that support and are more applicable in many cases.