As long as you have to ask this question, I assume you've never seen a course on algebraic geometry or advanced abstract algebra, so I'll try to explain by not using "fancy" words, here I go...
First of all, roots of polynomial equations are not "algebraic objects", in any book on abstract algebra we define polynomials and then we define polynomial functions (try not to think of graphs of functions here) and then we define zeros of polynomial functions, that is, roots of a polynomial equation. No mention of any geometry here!
BUT... there is a specific context where this extract from Wikipedia is right, but this is "just" a matter of interpretation.
Consider for example the following polynomials:
$$(i)\ x^2+y^2-1 = 0$$
$$(ii)\ y-x^2 = 0$$
$$(iii)\ y^2-x^3+x = 0$$
When we try to find solutions for these equations in the set $\mathbb{R}^2$, that is, the real plane, the solutions have a "nice" format, they are very "geometric", for example, the set of solutions of $(i)$ is a circle, $(ii)$ is a parabola and $(iii)$ is one example of what we call elliptic curves.
Now consider the following polynomials:
$$(I)\ x^2+y^2=0$$
$$(II)\ x^2+y^2+1=0$$
What about now ? The only "point" in $\mathbb{R}^2$ that is solution of $(I)$ is the origin, $(0,0)$, even worse is the case for $(II)$, there is no solution in $\mathbb{R}^2$.
But if we search for solutions in the complex plane $\mathbb{C}^2$ we find that:
$$(I)\ x^2+y^2=0 \Rightarrow (x+iy)(x-iy)=0$$
$$(II)\ x^2+y^2+1=0 \Rightarrow u = ix, \ v = iy \Rightarrow u^2+v^2-1=0$$
now have infinitely many solutions and we "restore" the geometry of the solutions set, for example $(I)$ is the union of two lines and $(II)$ is a circle!
So by using complex numbers we always (by the fundamental theorem of algebra) get "nice" sets of solutions for polynomial equations, and in this sense zero of polynomials are geometric objects.
Final remark: Above I said that this is "just" a matter of interpretation, but this slightly change in the view point give us one of the most glorious and exciting branch of mathematics, this give us algebraic geometry.