As my Group theory professor told me, giving a root of $x^2-2=0$ the name $\sqrt 2$ doesn't magically "solve" the equation. Calling a number $\sqrt 2$ is just saying "it solves $x^2-2=0$". You haven't gained any new information. Similarly, if $p(x)$ is an irreducible polynomial, then simply writing down $p(x)=0$ could be considered to be "solving" the equation, in the sense that, in a certain technical sense, the fact that $x$ solves that equation contains all relevant information about $x$.
Solving by radicals means that we wish to express roots of a polynomial in terms of the roots of a particular family of polynomials $x^2-a$. One reason to do this is that it means that we can numerically approximate those roots using methods to numerically approximate roots of that simpler family of polynomials. This is a common strategy in mathematics. For example, in trigonometry we try to express all quantities using $\cos$ and $\sin$. In a sense, this doesn't mean we've "calculated" those quantities, since $\cos$ and $\sin$ are just names given to particular ratios of line segments, but it does mean that we can reduce the problem of approximating various quantities to just approximating $\cos$ and $\sin$.
The Bring radical represents the applicaiton of this strategy to quintics. We choose a particular one-dimensional (in the vector space sense) family of polynomials and invent a special symbol to refer to a particular root of a polynomial in that family. It can be shown that in this way we can express the roots of any quintic.