Solubility of Organic Compound confusion
There are two regions of bond energy to look at here:


The bonds IN a molecule: a carbon-hydrogen bond has about $\pu{100
    kcal/mol}$ of stabilization. An oxygen-hydrogen bond is a little
stronger, about $\pu{110 kcal/mol}$. Stronger means HIGHER in 
stabilization energy; but we draw diagrams where the stronger bonds are 
designated by lines that are LOWER on the diagram. We hope no confusion
results!
The bonds BETWEEN molecules of a pure material or a mixture. These
(much weaker) bonds between molecules of a substance can be
estimated by heats of fusion or vaporization. The heat/enthalpy of
fusion of $\ce{H2O}$ is $\pu{1.435 kcal/mol}$, and for $\ce{CH4}$, $\pu{0.223 
kcal/mol}$. Hydrogen bonding (not the regular bond to hydrogen, but the polar
bonding between molecules) falls in this region of energy and is
frequently encountered when examining solutions.


The bonding between water molecules is strong enough to resist separation by methane molecules because the gain could be no more than the stabilization in methane ($\pu{0.223 kcal/mol}$), but the loss would be as much as $\pu{1.435 kcal/mol}$. The first sentence in the question refers to this region of bond energies, not to the much higher energy region (~ $\pu{100 kcal/mol}$) that would be extensively covered in a chapter on bond enthalpy.