Why are alkenes and alkynes more reactive?
First you must know the basics
The most basic explanation (that you would hear in chem. 101) is that alkynes are more reduced (less saturated) than alkanes (and alkenes as well) so there is more potential for hydrogenation (addition of hydrogen) and more potential energy to be released from such a reaction. The more bonds formed, the more energy released. This is the reason that fats contain more energy than carbohydrates... both of these molecules have alkane backbones, but the basic idea is the same, since fats are less oxidized and therefore higher in potential energy.
It's not a question of more bonds, but type of bonds. Alkanes are formed through sigma bonding. Alkynes demonstrate two pi bonds, which are extremely electron rich. When you have a large concentration of electrons, it's going to want to balance the charge so it is extremely nucleophilic. Pi bonds are much weaker than sigma bonds and hence are easily broken.
Sources my textbook