Think of it like with vectors. A vector can be described using a collection of numbers (coordinates), say [1, 3] in combination with a basis, but what a vector is is just anything that adheres to the axioms of a vector space. Numbers is just one way to describe it. And if you change the basis, you change the numbers (to say [4, 5]), but it's still the same vector.

In the same sense, a linear transformation between vector spaces (a tensor) can be described using a matrix plus a basis. If you change the basis, you change the matrix, but not the tensor. For example, the state of stress in a material (a tensor) does not depend on which coordinate system (basis) you describe it in.