Why does the Higgs field have less energy when it's non-zero than when it's zero? Why  does the Higgs field have less energy when it's non-zero than when it's zero? There are references to this question on the site, but they are too heavy going for me for a while yet. Anybody want to try a non math (or minimum math ) answer/analogy?
The basis for this question is p35 of Carroll's pop-sci book "The Particle at the End of the Universe" , in which there is a diagram comparing the resting value of fields such as electrons, quarks etc. as zero but the Higgs resting value as some non-zero value. I do understand that the Higgs field is not derived from a gauge theory, and that, afaik, all other fields are.
I simply want an as physical as possible, intuitive picture, rather than maths basis, for the discrepancy between the resting values of all other fields and that of the Higgs. 
I self study, so whilst I can follow most of the maths up involved in QM to the Dirac equation level, my knowledge is (very) patchy in parts and sometimes its easier, as a QFT newbie, to get a analogy before diving into the underlying maths. I do fully understand that a physical picture is often misleading, or impossible. If it's not possible to give a physical explanation/picture, then that's perfectly OK and  I will keep digging until I understand it on a maths level.
 A: From the form of the Higgs potential (which is quartic, the famous Mexican hat) you can see that for $\Re \phi =0$ as well as for $\Im \phi=0$ (the real and imaginary parts of the Higgs field), it is sitting on the unstable top of the hat. Thus, a small perturbation would lead it away from $0$. Since the potential has a smaller value away from zero so does the energy. Take a look at this picture for more intuition.
A: This is more of an extended comment, because I don't know the answer, but hopefully this will be useful.
Any field with a non-zero spin must have to have a vacuum expectation value (VEV) of zero, because any other value would break Lorentz invariance. So spin 0 bosons like the Higgs are the only ones that can have non-zero VEVs.
As for the form of the Higgs potential, the one in the standard model is the only one that is renormalisable. So assuming the Higgs is correctly described by the standard model it has to have a potential of this form.
But none of this explains why the Higgs VEV is non-zero or what the physical interpretation is. I don't know of an explanation for this except that, well, that's the way the universe is.
There is a cliché you'll come across in quantum mechanics:

anything that isn't forbidden is compulsory

and it's not forbidden for the Higgs to have a non-zero VEV. So at the end of the day maybe the Higgs has a non-zero VEV because it can.
