Robert gave a good answer for your specific question. In general when asked to find the domain of a function, we are interested in values of x where we get a real valued answer. That means the inside of a radical expression must be non negative. So they can be 0 or 4 but not -3 or -14 for example inside of the radical, since the solution to negative radicals are complex and this outside of our domain.
Therefore the expression sqrt(x+5), the x+5 must be positive or zero, so the least number x can be is -5. When we consider both expressions, both must follow this rule which is why we get that x must be greater than or equal to -1.