Grassmann's original motivations came from mechanics and geometry. Here is what he wrote in the foreword to his 1844 book "Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (Linear Extension Theory, a new branch of mathematics)":
"While I was pursuing the concept of product in geometry as it had been established by my father, I concluded that not only rectangles but also parallelograms in general may be regarded as products of an adjacent pair of their sides, provided one again interprets the product, not as the product of their lengths, but as that of the two displacements with their directions taken into account. When I combined this concept of the product with that previously established for the sum, the most striking harmony resulted; thus whether I multiplied the sum (in the sense just given) of two displacements by a third displacement lying in the same plane, or the individual terms by the same displacement and added the products with due regard for their positive and negative values, the same result obtained, and must always obtain."
More can be found at http://www-history.mcs.st-and.ac.uk/Extras/Grassmann_1844.html
There is a very accessible account in the talk "Grassmann, Geometry and mechanics" by John Browne: