How is it possible that a catalyst without taking part in a chemical reaction increases the speed of the chemical reaction?
Because it provides an alternative reaction pathway, lowering the activation energy.
Note that catalysts are not magic. A catalyst has NO effect whatsoever on the thermodynamics of the reaction. A reaction that is endothermic or exothermic will remain endothermic or exothermic by the same magnitude whether the reaction is catalyzed or not.
Catalysts, can, however, alter the activation energy, the energy threshold that a molecule requires for successful reaction. A catalyst may combine with the substrate (so the reactants are called), and form an activated complex; that is a species for which reaction is much more facile. Why? Because the activation energy for the given reaction has been REDUCED, and a greater proportion of the reactant molecules possess the required energy. Once the chemical transformation takes place, the catalyst detaches from the product, and may catalyze another transformation. (You should look at energy profiles on the web for catalyzed versus uncatalyzed reactions, and you will appreciate the difference in activation energies.) Enzymes are the classic examples of catalysts. Note here that catalysts do in fact take part in the chemical reaction, but ideally they do so WITHOUT being chemically altered, and they detach from the product after reaction occurs.
There are class of catalysts that impede reactivity by increasing the activation energy, a so-called inhibitory catalyst. How do you think these would affect the reaction?