I'm not a teacher, but here's my intuition for algebraic numbers.

You start with the natural numbers. You introduce the integers as all solutions x to x+y=z for any and all natural numbers y and z, which also completely solves the equation x+w=0 for all integers w. Rationals come from the similar equations x*y=z and px+q=0 for integers y,z and rationals p,q. The constructible numbers are the solutions to quadratic equations with integer coefficients. Each extension has been to solve some polynomial with integer coefficients, firstly linear polynomials, and then quadratic. Algebraic numbers are then the natural conclusion when extending to polynomials of arbitrary degree with integer coefficients.

Again, I am not a teacher. This might not be the best way to teach these students, but it's how I self-taught myself the intuition for algebraic numbers.