The simplest $SU(5)$ GUT Higgs transforms as ${\bf 10}$ under the gauge group, an antisymmetric tensor $5\times 4/2\times 1$ with two indices of the same kind (without complex conjugation). The 2-dimensional representation of $SU(2)$ has an antisymmetric invariant $\epsilon_{ab}$ and if you extend this antisymmetric tensor to 5-valued indices of $SU(5)$ and only make the $ab=45$ component nonzero, it will break the $SU(5)$ down to $SU(2)$ rotating $45$ and $SU(3)$ rotating the remaining $123$.
One could a priori think about other representations, for example ${\bf 15}$, the symmetric tensor with two indices $5\times 6/2\times 1$. It passes the basic test: You may imagine it determines a bilinear form on the 5-dimensional fundamental representation that has a different coefficient for the group of 3 basis vectors and different for the remaining 2 basis vectors among the 5, so something that tells you
$$ds^2 = A(da^2+db^2+dc^2)+B(dd^2 +de^2) $$
where $A,B$ are different complex coefficients and $(a,b,c,d,e)$ is a complex 5-dimensional "vector" in the fundamental representation. It's easy to see that distinct values of $A,B$ break the rotational symmetry $SU(5)$ between all five $(a,b,c,d,e)$ to $SU(3)\times SU(2)$ between $a,b,c$ and $d,e$ separately.
It's hard to write realistic potentials for this one – and moreover, the hypercharge $U(1)$ which should be composed of the $U(1)$ factors in the $U(2)$, $U(3)$ subgroups – won't arise properly (the bilinear form above isn't invariant under any such $U(1)$) – but there exist other, larger representations for the Higgs in $SU(5)$ that can potentially do the breaking job.
In $SO(10)$ gauge theories, one usually needs a 16-dimensional representation to do the Higgsing to $SU(5)$. The $SU(5)$ is the subgroup preserved by a single chiral spinor. There may also be a 126-dimensional Higgs multiplet to do similar things (antisymmetric, self-dual, with 5 indices) but I don't want to list all group theory used in grand unification here.
In string theory, the breaking of the GUT gauge group often proceeds by non-field-theoretical mechanisms such as fluxes and the Wilson lines around some cycles in the compactified dimensions. The Wilson line is a monodromy, an element of the original unbroken gauge group, and the gauge subgroup that commutes with the monodromy remains unbroken. It has some advantages because the required Higgs fields in GUT theories (and their potentials) may be rather messy and moreover, the stringy approach may justify more structured Yukawa couplings for various quarks and leptons which is probably needed.
GUT theories have their characteristic energy scale, the GUT scale, so all massive things such as the $X,Y$ new gauge bosons as well as the new GUT Higgses are naturally this heavy, near $10^{16}\,{\rm GeV}$. There are other ways aside from the proton decay constraints to derive this energy scale - it's the scale at which properly normalized three Standard Model gauge couplings approximately unify (almost exactly when supersymmetry is added).
So before one answers your question, it must be reverted. The right question is why the other fields (and dimensionful parameters) are so immensely light relatively to the GUT scale. Because most of them are derived from the electroweak Higgs mass to one way or another (gluon is formally massless although it's confined at the QCD scale, and one explains the QCD scale as the scale at which the slowly logarithmically running QCD coupling just grows to 1 if we run from a reasonable value near the GUT scale), this question really asks why the electroweak Higgs boson is so much lighter than the GUT scale. This question is known as the hierarchy problem and it's been the primary mystery that was driving much of the work in phenomenology and model building although the LHC, by its seeing nothing new, is increasingly suggesting that there may be no "nice answer" to this puzzle at all.
I think you have understood it almost well.
The masses do not change, they are what they are; at least at colliders. At high energy, it is true that the impact of masses and, more generally, of any soft term, becomes negligible. The theory for $E\gg v$ becomes very well described by a theory that respects the whole symmetry group.
Notice that to do so consistently in a theory of massive spin $-1$, you have to introduce the Higgs field as well at energies above the symmetry breaking scale. For the early universe, the story is slightly different because you are not in the Fock-like vacuum, and there are actual phase transitions (controlled by temperature and pressure) back to the symmetric phase where in fact the gauge bosons are massless (except perhaps for a thermal mass, not sure about it).
EDIT
I'd like to edit a little further about the common misconception that above the symmetry breaking scale gauge bosons become massless. I am going to give you an explicit calculation for a simple toy mode: a $U(1)$ broken spontaneously by a charged Higgs field $\phi$ that picks vev $\langle\phi\rangle=v$. In this theory we also add two dirac fields $\psi$ and $\Psi$ with $m_\psi\ll m_\Psi$. In fact, I will take the limit $m_\psi\rightarrow 0$ in the following just for simplicity of the formulae. Let's imagine now to have a $\psi^{+}$ $\psi^-$ machine and increase the energy in the center of mass so that we can produce on-shell $\Psi^{+}$ $\Psi^{-}$ pairs via the exchange in s-channel of the massive gauge boson $A_\mu$. In the limit of $m_\psi\rightarrow 0$ the total cross-section for $\psi^-\psi^+\rightarrow \Psi^-\Psi^+$ is given (at tree-level) by
$$
\sigma_{tot}(E)=\frac{16\alpha^2 \pi}{3(4E^2-M^2)^2}\sqrt{1-\frac{m_\Psi^2}{E^2}}\left(E^2+\frac{1}{2}m_\Psi^2\right)
$$
where $M=gv$, the $A_\mu$-mass, is given in terms of the $U(1)$ charge $g$ of the Higgs field. In this formula $\alpha=q^2/(4\pi)$ where $\pm q$ are the charges of $\psi$ and $\Psi$.
Let's increase the energy of the scattering $E$, well passed all mass scales in the problem, including $M$
$$
\sigma_{tot}(E\gg m_{i})=\frac{\pi\alpha^2}{3E^2}\left(1+\frac{M^2}{2E^2}+O(m_i^2/E^4)\right)
$$
Now, the leading term in this formula is what you would get for a massless gauge boson, and as you can see it gets correction from the masses which are more irrelevant as $m_i/E$ is taken smaller and smaller by incrising the energy of the scattering.
Now, this is a toy model but it shows the point: even for a realistic situation, say with a GUT group like $SU(5)$, if you scatter multiplets of $SU(5)$ at energy well above the unification scale, the masses of the gauge bosons will correct the result obtained by scattering massless gauge bosons only by $M/E$ to some power.
Best Answer
An experimentalist's answer,
Our observations tell us that baryon and lepton number are conserved, within the accuracies of our experiments and observations. This means we have chosen as a standard model SU(3)xSU(2)xU(1) because in the group structure of the possible representations of all the quantum numbers assigned to the particles and resonances we know, there is no exchange particle that could change a proton to something else and no calculable Feynman diagram like:
This means that within the standard model the probability for a proton to decay is zero with any precision in the perturbative expansions which we calculate crossections with.
Extending the standard model, i.e.stating that it could be embedded in a higher group's representations, as an SU(5) model, with the standard model embedded, introduces new particles that can be exchanged, which carry lepton and quark numbers such that their exchange allows a proton decay. In this specific model these diagrams have been calculated and give a lifetime of order of 10^36 years.
The experimental limits of the half life are close to 10^34 years.
So it is not that
They are fundamental in SU(5). It is the group structure that is extended, and new exchange paths that allow Feynman diagrams that lead the (non fundamental) proton to decay. This larger structure also introduces new particles, like leptoquarks. To discover leptoquarks is an objective for the new lepton collider under consideration.
This paragraph summarizes :
This paragraph explains the "stunned":
I suppose the book was written before the later calculations that recalculated the limit to 10^36years.
The reason we are not satisfied with the standard model and are trying extensions to higher groups and symmetries comes from the experimental observation that the coupling constants of the three interactions in the standard model tend towards the same region as the energies get larger. The theoretical urge to unite gravity to the other three forces which is supported by the cosmological observations to date are another strong impetus. But this is another stunning story :).
It is string theories that are trying to change our concept of "particle with specific quantum numbers" to "fundamental vibrating strings with specific vibrational levels". The future will tell.