I think these are two different questions:
- Why care about radicals?
- Why try to solve polynomial equations in terms of radicals?

For the first question:

Taking Nth powers is a fairly basic operation, which occurs all the time in mathematics.
Taking Nth roots is simply the inverse operation, so it is fairly natural to be interested in it/having to deal with it.

For the second question:

Let’s pretend for a moment that we didn’t know how the quadratic formula looked like.
Could we nevertheless say anything about it?

The quadratic formula is supposed to give us the solutions to the equation a x^2 + b x + c = 0.
A special case of this general quadratic equation is x^2 - p = 0.
There are two ways of solving this specialized equation:
either by taking a square root, giving us the two solutions ±√p, or by using the general quadratic formula (with a = 1, b = 0, c = -p).
Both of these approaches need to give us the same results, since they are both correct.

This tells us that if we simplify the quadratic formula with a = 1, b = 0, c = -p, then a square root needs to appear.
How can this happen?
Well, the most basic guess is that the quadratic formula contained at least one square root to begin with.

Looking at the actual quadratic formula tells us that this guess is correct:
the formula uses the four basic arithmetic operations (addition, subtraction, multiplication, division) and a square root.

We can repeat the same thought experiment for cubic equations, and we find that the cubic formula should probably contain third roots.
Looking up the formula confirms this suspicion.
However, it should be noted that the cubic equation does not only contain third roots, but also square roots.

The situation for the quartic equation is similar:
we suspect that the quartic formula contains fourth roots.
And thanks to our experience with the cubic formula, we may also suspect that the quartic formula contains third roots and square roots.
Looking up the formula, we see that it contains both third roots and square roots, but not (directly) any fourth roots.
(Our original idea breaks down a bit because fourth roots can be expressed as iterated square roots.
This makes it possible that the general quartic formula does not contain fourth roots, even though its simplified version will contain them.)

So what about a general polynomial equations of degree N >= 5?
Our original observation tells us that a solution formula needs to contain some sort of operation(s) that, when the formula is applied to certain special cases, gives us Nth roots.
Just as before, the most basic guess is that the formula will contain Kth roots, and the previous examples suggest that one should expect K = 2, ..., N to occur.

Summary:
To find a formula for polynomials equations of degree N >= 2, we are forced to use additional operations apart from the four basic arithmetic operations.
In certain special cases, these additional operations need to simplify to roots.
This suggests using roots in the formula, and the cases N = 2, 3, 4 support this idea.

Heuristically speaking, we are not trying to use roots because we want to, but because they seem to be the bare minimum required to even hope of finding a formula.