So, yeah, I'm afraid I'm not able to give a good layperson's explanation for the unitarity bound on the Standard Model higgs mass. The basic idea is that the probabilities of certain interactions (specifically, vector boson scattering) depend on the higgs mass, and if the higgs mass is too large, those probabilities end up greater than 100%. We interpret such a nonsensical result as either setting an upper limit on the higgs mass in the framework of the Standard Model, or indicating the scale at which some new physics (such as supersymmetry or technicolor) must come into play to make the Standard Model an inappropriate framework at that scale.
I don't consider that satisfying because I have no non-technical way of explaining why the problems arise. I'll briefly sketch out the outline of the technical derivation, but I wouldn't expect anyone other than folks who've done some graduate quantum field theory to be able to follow it, no matter how much detail I give.
We start off analyzing vector boson scattering (WW \to WW, W^+W^- \to ZZ, etc) using the goldstone boson equivalence theorem. This roughly states that at high energies (much greater than the masses of the W and Z), the longitudinal components of the W and Z behave like massless scalar goldstone bosons. This is an aspect of the common saying that the W and Z each "eat" a goldstone boson degree of freedom to become massive through spontaneous symmetry breaking. This leads to the result (which also follows more generally from considering polarization factors) that at high energies, the amplitude for longitudinal gauge boson scattering dominates the amplitude for transverse gauge boson scattering.
For example, at high energy the amplitude for longitudinal W^+_LW^-_L \to W^+_LW^-_L scattering is
\mathcal A = -\frac{m_h^2}{v^2}\left(\frac{s}{s - m_h^2} + \frac{t}{t - m_h^2}\right),
where
v = \left(\frac{G_F \sqrt 2}{(\hbar c)^3}\right)^{-1/2} = 246 {\rm\ GeV}
is a constant known accurately from precision electroweak experiments, and m_h is the mass of the higgs.
To connect to unitarity, we break that amplitude up into a partial-wave decomposition,
\mathcal A = 16\pi\sum_{l = 0}^{\infty}(2l + 1)P_l(\cos\theta)a_l.
The optical theorem, which must hold if unitarity is not violated, gives bounds on the coefficients a_l, in particular
|\Re(a_l)| \le \frac{1}{2},
which follows from
\Im(a_l) = |a_l|^2 = \Re(a_l)^2 + \Im(a_l)^2 \Rightarrow \Re(a_l)^2 = \Im(a_l)(1 - \Im(a_l)) \le \frac{1}{4}.
We can find an expression for a_0 by integrating over
\int_{-1}^1 d\cos\theta \to \int_{-s}^0 dt
a_0 = \frac{1}{16\pi s}\int_{-s}^0\mathcal A dt = -\frac{m_h^2}{16\pi v^2}\left[2 + \frac{m_h^2}{s - m_h^2} - \frac{m_h^2}{s}\log\left(1 + \frac{s}{m_h^2}\right)\right]
The high-energy limit m_h^2 \ll s gives
\frac{m_h^2}{8\pi v^2} \le \frac{1}{2} \Rightarrow m_h \le 870 {\rm\ GeV},
while if we try to get rid of the higgs, which we can do by taking its mass to infinity, m_h \to \infty, m_h^2 \gg s, we find
\frac{s}{32\pi v^2} \le \frac{1}{2} \Rightarrow \sqrt s \sim 1.8 {\rm\ TeV}
as the maximum scale at which new physics must appear to preserve unitarity (conservation of probability).
As a final note, if I recall correctly, you can get a slightly lower bound of
m_h \le 710 {\rm\ GeV}
by performing a similar calculation for W^+_LW^-_L \to Z_LZ_L scattering, which is more complicated since it's inelastic and has identical particles in the final state.
