How to define equivalents here
Number of equivalents and normality are out-of-date concepts in chemistry. For acids, they are just like number of mols and molar concentration (or molarity), with an extra factor indicating how many hydrogens can be effectively ionized in solution. Because acetic acid ($\ce{CH3COOH}$) is monoprotic, you can just exchange the terms by number of mols and molarity and you are good.


  Why is normality being decreased here?


Some of the acetic acid adsorbs - meaning it bonds - in the charcoal. So the concentration in solution is decreased. Pay attention to the fact that the charcoal remains solid and does not dissolve.


  Why are they taking volume to be same when it generally increases when we add solutes?


As I wrote above, the charcoal stays solid, so it has no effect on the solution volume. The loss of acetid acid in solution might have a slight effect on volume, but that is mostly negligible.