Geometrical meaning of Grassmann Algebra

You could consider going back to the source! There is a good English translation of Grassmann's original work, which is all rooted in his geometric intuition for what is now called multilinear algebra and Grassmann algebras. Of course, you'll also have to suffer through a lot of metaphysical and theological mumbo-jumbo to get at the mathematics. But the mathematics is brilliant indeed. I have long thought that his examples were always more convincing to me than any of the modern texts -- although the modern texts have much clearer mathematical definitions! I really wish that modern texts were written with the mathematical clarity and rigor of 'now', but with the detours into motivation and intuition best seen in the classics (i.e. mathematical papers from the early 1700s to the early 1920s).

I'm going to vote for Guillemin and Pollack's chapter of "Differential Topology".

Basically, a k-form should be something that you can integrate over k-dimensional submanifolds. And it shouldn't matter how you parametrize them. That means that there should be determinants baked in to the definition, since those measure how the volume changes when you change coordinates. The determinant of a matrix negates when you switch two rows, so a k-form should be antisymmetric this same way. That's pretty much it.

For a brief explanation of the geometric meaning of exterior product, interior product of a k-form and l-vector, Hodge dual etc. see my answer here: When to pick a basis?

The best reference for this stuff is Bourbaki, Algebra, Chapter 3.