Inflation resolves the particle horizon problem with CMB. On page 27 of the book they calculate the particle horizon comoving radius at time of last scattering to be 90 Mpc/h. On the same page they calculate the comoving radius to the last scattering surface (a sphere centered on us from which the CMB we observe today was emitted) to be 5820 Mpc/h. The very high homogeneity of CMB suggests two diametrically opposite points on the last scattering sphere should have been in causal contact at time of last scattering, yet they are not within eachothers particle horizons at that time because their particle horizons were much smaller than the distance between them: 90 << 5820.
The explanation that inflation provides is that the comoving particle horizon radius (estimated as 1/Ha) before inflation was much bigger allowing for the two points to be in casual contact and homogenize with each other. Later during inflation 1/Ha shrinks. After end of inflation till the last scattering it expands again but not enough to be equal to the comoving radius of the last scattering surface. That creates the illusion that the two points have never been in causal contact but they were. That is shown on fig 3.2 in the book.
Now to answer your question. Shrinking of the particle horizon 1/Ha in comoving coordinates simply means that the particle horizon in 'physical' coordinates, 1/H, increases slower relative to the scale factor a. During inflation the scale factor increases exponentially while 1/H is approximately constant and the ratio 1/Ha shrinks exponentially. That means during inflation matter that was homogenized in the past is flowing out of the physical particle horizon radius 1/H, because the horizon is not expanding as fast as the universe. Matter flowing out of particle horizon in physical coordinates shows as shrinking of particle horizon in comoving coordinates because in comoving coordinates matter appears frozen so the radius must shrink for the matter to get outside of it. That creates at last scattering the apparent paradox of two points outside of eachothers physical horizon which are nevertheless at the same temperature. The explanation provided by inflation is that they were inside eachothers horizon before inflation.
