That abstract approach tends to be how mathematicians view the tensor product (there is also a categorical construction), but I don't find it very helpful for u

That abstract approach tends to be how mathematicians view the tensor product (there is also a categorical construction), but I don't find it very helpful for understanding what tensors do, or why they are useful in physics. With the "multilinear map" definition, taking the tensor product T of tensors U and V just means evaluating U and V respectively on the arguments of T and multiplying their outputs. Extend this definition by linearity and you have the tensor product of spaces of tensors.