Perhaps this will help. From the Pauli exclusion principle, we know that fermion wave functions have to be "odd" or "anti-symmetric". Thus, for example if \(\psi(x,y)\) describes two fermions, then one must have \(\psi(x,y)=-\psi(y,x)\) under the exchange of coordinates of two particles. Moving to quantum field theory, we want to be able to describe N-fermion states, and we also want to be able to create and destroy them: e.g. destroy a fermion moving in one direction, and create one moving in another direction. Alternately, maybe destroy an electron-positron pair, and create a pair of photons. This forces the use of the concept of Fock space, and the use of creation/annhilation operators. The entire process of setting this up is called "second quantization".
The issue is that the two operators you wrote down are misleadingly over-simplified: they are not enough to create an N-particle state; they are not enough to create a state where the N particles have distinct, different momenta, or maybe distinct, different positions. To do it properly, you have to work with ladder operators that do this. Now when you do so, and you create an N-fermion state, it must be completely anti-symmetric under the interchange of particle indexes: exchanging any pair must flip the sign on the wave function: this is the Pauli exclusion principle at work -- this is what it demands.
The \(b\) and \(b^\dagger\) operators you wrote are just enough to construct exactly one fermion in one specific momentum-state (or position-state, etc). That is why they square to zero -- you can't have two fermions in the same state. But there is no clue, in the simplistic way that they are written, that the resulting particle states will be anti-symmetric. (Caution: the ladder operators themselves are not Grassmann variables. The Grassmann variables discussed here are to be used exclusively for the wave functions. What's more, the Grassman numbers themselves are not spinors! This is a common source of confusion, especially if you wish to get \(\overline{\psi}\psi\)as a non-vanishing quantity; see the end-note.).
The Grassmann variables are a book-keeping device that helps you keep track of the sign, during any calculations. Swap two of them, and the sign changes. You don't have to use them, but if you don't you will probably make more errors. The anti-commuting might seem strange, but anti-commutation is very common in differential geometry: two differential forms anti-commute. The correct way to think of Grassmann numbers (see wikipedia) is to pretend they are differential forms on some manifold, but then throw away/forget the manifold, and throw-away/forget that they are derivatives: all you have left is an abstract algebraic device that anti-commutes, and does nothing more. Equivalently, you can pretend that they are actual fermion wave-functions, and then completely ignore/forget that they have a position, a momentum, a spin, or even that they are wave-functions at all -- and keep only the fact that they anti-commute. That's it. End of story.
Well, almost end-of-story. Grassmann variables would be boring and almost useless if that was the end-of-story. From this point on, a vast number of interesting, surprising and outright amazing "accidents" happen, all involving Grassmann numbers in some way or another. Most immediately, in your case, they allow you to write exact expressions for Feynmann path integrals involving fermions as a determinant. Now, determinants have all sorts of amazing properties of their own, involving operator algebras, the det=exp-trace-log relation, Poincare duality, K-theory, the Atiyah-Singer index theorem which relates fermions to topological twists of boson fields -- for example, the proton, at low energies, is accurately modeled by the "Skyrmion", which is a soliton in the pion field -- the Skyrmion gets the axial vector current of a proton just about right, it gets the proton radius just about right, etc. Yet we know protons are made of quarks, and so we have to somehow "rotate" those quark fields "into pions", and accomplish that rotation in a consistent fashion. This is just scratching the surface, though. There are too many neat directions one can go in: rotations lead to spin lead to Lie groups, lead to covering groups; the covering groups of rotations form a Postnikov tower, consisting of the rotation group, the spin group, the string group, the 5-brane group ... now, this is just/only "pure math", no physics involved -- just fiddling with Eilenberg-MacLane spaces. And yet the pure math leads directly and unavoidable to vocabulary like "string" and "brane" which are heavily used in theoretical physics. How can that possibly be? Hmmm.
The point is that Grassmann numbers need to be viewed as a handy-dandy calculational device that neatly captures and balls up the Pauli exclusion principle. The Grassman numbers are just plain, ordinary elements of the exterior algebra, which is central in mathematics. You will see the exterior algebra over and over again, in both math and physics, so you better get very comfortable with it. More generally, get comfortable with what an "algebra over a field" is -- its just a vector space, where you can multiply vectors. But algebras are pervasive, and Grassmann is just one example. There are many others, important to physics: Clifford algebras, Jordan algeras, Lie algebras, etc. If you say to yourself, "its just an algebra with weirdo multiplication" you can get through an awful lot of material.
Caution: The algebra generated by the ladder operators is not the Grassmann algebra, it is a Clifford algebra, and not just any Clifford algebra, but a Clifford algebra over the complex numbers, and specifically, a certain subspace of that, the spin algebra. The correct construction of this can be subtle and confusing. so here it is, copied from page 69 of Jurgen Jost's book "Riemannian Geometry and Geometric Analysis" : Let \(e_i\) be the ordinary real basis vectors of a real vector space \(V\). Construct a tensor algebra, and from that by taking the appropriate quotient, a Clifford algebra, so that \(e^2_i=-1\) and \(e_ie_j+e_je_i=0\) So far, this is just a real algebra (not complex). These wacky looking algebraic relations may look .. strange, but they are perfectly well-defined and arise purely as the result of taking a quotient on the tensor algebra. The tensor algebra seems like a beguilingly simple object; its not, it has subtlety. Likewise, so is quotienting. Anyway, the \(e_i\) really are meant to be the basis vectors of a real vector space, here. Next, suppose that \(V\) is even-dimensional and consider the vector space \(V\otimes\mathbb{C}\) and specifically a subspace \(W \) that is spanned by the vectors \(\eta_j =(e_{2j-1}-ie_{2j})/\sqrt{2}\) Taking the plain-old ordinary scalar product of vectors \(\langle,\rangle\) you can directly verify that \(\langle \eta_i,\eta_j\rangle=0\) and that \(\langle\eta_i,\overline{\eta_j}\rangle=\delta_{ij}\) where the overline denotes complex conjugation. This is exactly the algebra of the ladder operators that you wanted, and it was explicitly constructed as the subspace of a certain Clifford algebra. Technically, one writes \(V\otimes\mathbb{C}=W\oplus \overline{W}\)and the exterior algebra \(\Lambda W\) of \(W\) is the spinor space -- that is, it is \(\Lambda W\) that is the Grassmann algebra that you are working with, in physics. That is, the wave function of a single fermion (Weyl spinor) belongs in the (vector space!) \(W\) and that for n of them belong to \(\Lambda^n W\) The ladder operators are then certain endomorphisms in \(\Lambda W \) and \(\Lambda \overline W\) and, as any endomorphism would do, inherit some of the algebraic properties of the space that they act on. Dirac fermions are direct sums of one Weyl spinor, and one anti-spinor (four numbers are used to represent a Dirac spinor, not two, and zitterbewegung means that two of them behave as an anti-particle.). All this can be quite subtle, confusing, and subject to a whole mess of numbing, confusing details -- the Spin group, the Pin group, the spin structures and spin manifolds, as is readily explored in the Jost book.(Wikipedia does not quite have enough detail, here) If you have the time, plowing through these is a worthwhile exercise: it will turn you into the kind of student that the professors hunt down and respect.
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