You can imagine multilinear functions as a local approximations to nonlinear functions that depend on several variables, where the approximation takes into account the the linearization of each variable independently. Just like when you zoom in on a nonlinear function of one vector it looks approximately linear, if you zoom in on a function of many vectors it looks approximately multilinear.

If a basis is chosen for each input vector space, then a multilinear function is completely defined by its action on all possible combinations of basis vectors. The results of applying the multilinear function to all combinations of basis vectors can be arranged into a multidimensional array of numbers, and this array can be considered as a representation of the multilinear function, with respect to a given basis.

If one changes the basis, obviously the entries in the multidimensional array representation will change, but in a predictable way. The exact way in which the entries of the array change when you change the base are known as "transformation rules". In many physics classes, boxes of numbers obeying these transformation rules are presented as the definition of a tensor, which is a perfectly legitimate definition, but can be jarring and unmotivated if the multilinear context that these rules come from is not explained.

Since fixing all but one input to a tensor yields a linear function in the remaining input, and since linear functions on a vector space can be identified with elements in the dual to that vector space, a tensor can be equally viewed as a multilinear function that takes one less than the original amount of vectors as input, and produces a (dual-)vector as output. This output can then be used as one of the inputs to another tensor that has an input spot for a dual vector. More generally, complicated networks can be constructed where various inputs to a tensor are reinterpreted as outputs, and then those outputs are used as inputs into other tensors in the network (see Penrose graphical notation).

If one attaches a different tensor to every point on a surface (or manifold), the collection of all of these tensors is a tensor field (just like if one attaches vectors to all points on a manifold one gets a vector field). A common special case is where the input vector spaces that form the domain of the tensor at a point are copies of the tangent space and cotangent space of the manifold at that point. In this case, a convenient bases to use are the tangent and co-tangent vectors at every point associated with some pre-existing coordinate charts for the manifold. If one changes the parameterization of the manifold, the coordinate charts will change, so the bases for the tensor field at every point will change, so the array representations of the tensors at each point will change (but in a predictable way..).

You can imagine multilinear functions as a local approximations to nonlinear functions that depend on several variables, where the approximation structurally takes into account the fact that there are several inputs from several spaces. Just like when you zoom in on a nonlinear function of one vector it looks approximately linear, if you zoom in on a function of many vectors it looks approximately multilinear.

If collections of basis vectors are chosen that span each input vector space, then a multilinear function is completely defined by its action on all possible combinations of basis vectors. The results of applying the multilinear function to all combinations of basis vectors can be arranged into a multidimensional array of numbers, and this array can be considered as a representation of the multilinear function, with respect to the given bases.

If one changes the bases, obviously the entries in the multidimensional array representation will change, but in a predictable way. The exact way in which the entries of the array change when you change the bases are known as "transformation rules". In many physics classes, boxes of numbers obeying these transformation rules are presented as the definition of a tensor, which is a perfectly legitimate definition, but can be jarring and unmotivated if the multilinear context that these rules come from is not explained.

Since fixing all but one input to a tensor yields a linear function in the remaining input, and since linear functions on a vector space can be identified with elements in the dual to that vector space, a tensor can be equally viewed as a multilinear function that takes one less than the original amount of vectors as input, and produces a (dual-)vector as output. This output can then be used as one of the inputs to another tensor that has an input spot for a vector in the dual space that was output. More generally, complicated networks can be constructed where various inputs to a tensor are reinterpreted as outputs, and then those outputs are used as inputs into other tensors in the network (see Penrose graphical notation).

If one attaches a different tensor to every point on a surface (or manifold), the collection of all of these tensors is a tensor field (just like if one attaches vectors to all points on a manifold one gets a vector field). A common special case is where the input vector spaces that form the domain of the tensor at a point are copies of the tangent space and cotangent space of the manifold at that point. In this case, convenient bases to use can be formed from the tangent (or co-tangent) vectors at every point associated with some pre-existing coordinate charts for the manifold. If one changes the parameterization of the manifold, the coordinate charts will change, so the bases for the tensor field at every point will change, so the array representations of the tensors at each point will change (but in a predictable way..).

Hope this helps with the intuition. This perspective was wrought slowly through years of struggling to understand tensors myself; this post is what I wish someone had told me at the start.

Hope this helps with the intuition. This perspective was forged slowly through years of fighting to understand tensors myself from first principles; going from utterly confused at the beginning, to having tensors feel natural and clear now. This post is what I wish someone had told me at the start.