These are $n$-tuples of reals, added componentwise and multiplied via the “exterior product.” They were introduced by Grassmann in 1844 as part of a brilliant attempt to construct a vector algebra in $n$-dimensional space. Grassmann’s style was far from simple and his approach was ahead of its time
To my understanding, differential forms is the reason why Grassman algebras are so famous. But they only appeared in 1899, so how did Grassman's interest in Grassman algebras arise right in 1844?
I have thought that maybe it is related to some differential geometry matters. Googling tells me that Gauss put forth "Disquisitiones generales circa superficies curvas" in 1827. So, maybe Grassman was influenced by this? I am not sure.