Explanation for bond lengths in trans-hexatriene
If you derive the π-type molecular orbitals of hexatriene, the three lower-energy MOs which are filled would look something like this (image from p 33 of Fleming's Molecular Orbitals and Organic Chemical Reactions, Reference Edition):



I suspect what Clayden is getting at is that in the second MO, there is some antibonding character between C3 and C4, whereas the C1/C2 and C5/C6 interaction is purely bonding.