Historical background of the approach to multilinear algebra
The subject itself has various roots going back to the mathematics of the nineteenth century, in what was then called tensor analysis, or the "tensor calculus[?] of tensor fields". It developed out of the use of tensors in differential geometry, general relativity, and many branches of applied mathematics. Around the middle of the 20th century the study of tensors was reformulated more abstractly. The Bourbaki group's treatise Multilinear Algebra was especially influential  in fact the term multilinear algebra was probably coined there.
One reason at the time was a new area of application, homological algebra. The development of algebraic topology during the the 1940s gave additional incentive for the development of a purely algebraic treatment of the tensor product. The computation of the homology groups of the product of two spaces involves the tensor product; but only in the simplest cases, such as a torus, is it directly calculated in that fashion. The topological phenomena were subtle enough to need better foundational concepts.
The material to organise was quite extensive, including also ideas going back to Hermann Grassmann, the ideas from the theory of differential forms that had led to De Rham cohomology, as well as more elementary ideas such as the wedge product that generalises the cross product.
The resulting rather severe writeup of the topic (by Bourbaki) entirely rejected one approach in vector calculus (the quaternion route, that is, in the general case, the relation with Lie groups). They instead applied a novel approach using category theory, with the Lie group approach viewed as a separate matter. Since this leads to a much cleaner treatment, there was probably no going back in purely mathematical terms.
Indeed what was done is almost precisely to explain that tensor spaces are the constructions required to reduce multilinear problems to linear problems. This purely algebraic attack conveys no geometric intuition.
Its benefit is that by reexpressing problems in terms of multilinear algebra, there is a clear and welldefined 'best solution': the constraints the solution exerts are exactly those you need in practice. In general there is no need to invoke any ad hoc construction, geometric idea, or recourse to coordinate systems. In the categorytheoretic jargon, everything is entirely natural.
Conclusion on the abstract approach
In principle the abstract approach can recover everything done via the traditional approach. In practice this may not seem so simple. On the other hand the notion of natural is consistent with the general covariance principle of general relativity. The latter deals with tensor fields (tensors varying from point to point on a manifold), but covariance asserts that the language of tensors is essential to the proper formulation of general relativity.
Some decades later the rather abstract view coming from category theory was tied up with the approach that had been developed in the 1930's by Hermann Weyl (in his celebrated and difficult book The Classical Groups). In a way this took the theory full cicle, connecting once more the content of old and new viewpoints.
Content of multilinear algebra
The content of multilinear algebra has changed much less than the presentation, down the years. Here are further pages centrally relevant to it:
And from the point of view of applications:
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