Difficulty in understanding why the Cartesian product of two circles is equivalent to a torus?

Question

I recently learned that the Cartesian product of two circles is topologically equivalent to a torus, and I have some difficult times in understanding this statement. Here comes a few of my confusions.

1. Suppose we are taking the Cartesian product of $\{(x_1,y_1):x_1^2+y_1^2=r_1^2\}$ and $\{(x_2,y_2):x_2^2+y_2^2=r_2^2\},$ do we get a torus? I understand the explicit and parametric formulas for torus, could we mathematically prove that the Cartesian product of the two circles and the torus are equivalent?
2. After taking Cartesian product of the two circles, we get a set of points in $\mathbb{R}^4,$ but isn't a torus an object in $\mathbb{R}^3?$ How could these two be equivalent?

I have learnt metric space, but don't have much background knowledge in topology. Hope this is something that I could understand. Many thanks for any help and references in advance!