This is just an archetypical nucleophilic substitution reaction. The Br of the alkyl bromide is replaced and a new sigma bond is formed from a lone pair nucleophile produced from the reaction between the alkyne and NH2-.
The alkyne's conjugate base is very basic and readily displaces good leaving groups.
You are not reducing the alkyne to an alkene in this reaction (removal of a pi bond).
Amide ion is typically added from a salt in aqueous ammonia (ammonia auto-ionizes to amide ion, much like water to H+ and HO-). NH2- (amide) is a lone pair nucleophile.
Since the alkyl bromide is attached to a primary carbon, the formation of a carbocation isn't probable. The C atom attached to Br is electrophilic because of the significant dipole moment in that sigma bond. The C atom is a sigma bond electrophile.
The C atom is electrophilic because C is relatively low in energy as bonding electrons are withdrawn through inductive forces from Br; in fact, when attached to a C atom, Br is commonly referred to as "bromide," or Br-, because of the greater occupancy of the bonding electrons around bromine relative to C. Thus, Br is almost like an ion. This is also due to the transition state of the step in the reaction sequence that features the removal of Br as a nucleofuge, or leaving group, and the formation of the new sigma bond between C and the lone pair nucleophile, in which the C—Br bond is partially broken and bonding electrons are "sucked" towards Br.
Alkynes can be reduced to alkenes using ammonia, but a alkali metal (such as Na or Li) must be present. These are called dissolving metal reductions.