Can someone briefly explain for me, why each of the six assumptions is needed in order to compute the OLS estimator? I found only about multicollinearity—that if it exists we cannot invert (X’X) matrix and in turn estimate overall estimator. What about the others (e.g., linearity, zero mean errors, etc.)?
You can always compute the OLS estimator, apart from the case when you have perfect multicollinearity. In this case, you do have perfect multilinear dependence in your X matrix. Consequently, the full rank assumption is not fulfilled and you cannot compute the OLS estimator, because of invertibility issues.
Technically, you do not need the other OLS assumptions to compute the OLS estimator. However, according to the Gauss–Markov theorem you need to fulfill the OLS assumption (clrm assumptions) in order for your estimator to be BLUE.
You can find an extensive discussion of the Gauss–Markov theorem and its mathematical derivation here:
Furthermore, if you are looking for an overview of the OLS assumption, i.e. how many there are, what they require and what happens if you violate the single OLS assumption may find an elaborate discussion here:
I hope that helps, cheers!