Is Supersymmetry really swapping fermions with bosons? I've been studying supersymmetry for the last few months, and while I can do some mathematics with the Wess-Zumino model (show the Lagrangian is invariant under a susy transformation, find the Noether charges, etc) I realise that I don't actually know what a supersymmetric transformation does.
An infinitesimal transformation of a spin 0 particle is proportional to a spin half particle $\delta\phi=\bar{\epsilon}\chi$, and vice versa, but I don't know what this means for the universe. 
Unlike other symmetries I know, the universe doesn't actually seem symmetric under susy. e.g. If the entire universe was translated in some direction, or CPT, then we wouldn't be able to notice. Yet if we swapped all the fermions and bosons, then my seat that I am sitting on would be made of higgs particles and I'd fall on the floor. 
What am I misunderstanding about supersymmetry? I've heard that it's only a symmetry of the equations, but what does that mean for our universe? Can a supersymmetric transformation actually happen?
 A: Yes, supersymmetry really swaps fermions and bosons and hence predicts that every particle has a corresponding superpartner of opposite nature (fermions have bosons and bosons have fermions as partners). Note that this does not mean that there is some physical process that "implements" this symmetry - symmetries are abstract properties of physical systems and there need not be any way to apply them to a system in practice (consider that e.g. time reversal symmetry also has no "implementation" since so far we can not reverse time).
As of yet there is no evidence that our universe is supersymmetric (much to the disappointment of some theorists, and to the delight of others), so this doesn't mean anything for our universe. But if it is supersymmetric, then it is to be expected that some mechanism makes many of the superpartners of the particles we know from the Standard Model very heavy, so that they cannot be observed with the energies our current colliders can produce.
A: 
Is Supersymmetry really swapping fermions with bosons?

I wouldn't say supersymmetry is really swapping fermions with bosons. It's saying there's a certain symmetry wherein each particle has a superpartner. A boson has a fermionic superpartner, and a fermion has a bosonic superpartner. For example supersymmetry says the electron has a superpartner called a selectron, which is a boson. 
However we've never seen a selectron, and people say it would have a much larger mass-energy than an electron. So you can't just swap the electron for a selectron. 

I've been studying supersymmetry for the last few months, and while I can do some mathematics with the Wess-Zumino model (show the Lagrangian is invariant under a susy transformation, find the Noether charges, etc) I realise that I don't actually know what a supersymmetric transformation does.

I don't think anybody will be able to tell you that, because contemporary physics doesn't give you a description of the electron, or the selectron. It doesn't even tell you what happens in pair production and annihilation. We can swap bosons into fermions in pair production, and we can swap fermions into bosons in annihilation. There's no description of what this transformation does. In similar vein there's no description of swapping an electron into a selectron. Which you can't do anyway because the mass-energy difference. 

An infinitesimal transformation of a spin 0 particle is proportional to a spin half particle $\delta\phi=\bar{\epsilon}\chi$, and vice versa, but I don't know what this means for the universe. 

It doesn't mean anything for the universe. If SUSY can't explain how a real-world transformation from a photon to an electron occurs in pair production, she hasn't got the foundation upon which to propose the selectron. A mathematical symmetry just isn't enough. Especially when it's broken.   

Unlike other symmetries I know, the universe doesn't actually seem symmetric under susy. e.g. If the entire universe was translated in some direction, or CPT, then we wouldn't be able to notice. Yet if we swapped all the fermions and bosons, then my seat that I am sitting on would be made of higgs particles and I'd fall on the floor. What am I misunderstanding about supersymmetry? 

Perhaps it's this: SUSY is only a hypothesis. 

I've heard that it's only a symmetry of the equations, but what does that mean for our universe? 

The equations might not be describing our universe? 

Can a supersymmetric transformation actually happen?

I don't know how. I think other transformations can happen. For example, the positron is a time-reversed electron because it has the opposite chirality, not because it's an electron travelling back through time. See this answer where I tried to describe it with a gif played backwards. But I have no concept of what a selectron is, so like I said, I don't know how a supersymmetric transformation can happen.     
