Random error is the accumulation of an infinite number of variables, and because the sum of an infinite number of variables follows a normal distribution according to the central limit theory, the error is assumed to follow a normal distribution...
As for the dependent variable, it is a linear combination of a variable that follows the normal distribution (error) and some constants (regression parameters and non-random independent variables)... According to the linear property of the normal distribution, any linear combination in terms of a variable that follows the normal distribution follows the normal distribution.