Why does $N{{H}_{3}}$ have a larger bond angle than $P{{H}_{3}}$ ?

Hint:The attraction and repulsion between the electrons inside the molecule is responsible for the bond angle present in the molecule.
The differences in the bond angles of different molecules are due to their own bond pairs and the lone pairs of the electrons.

Complete step-by-step answer:Let us learn the concept of bond angle in detail with respect to the molecules listed above;
Bond angle-
As we know from general mathematics, that we need at least two lines to form an angle between them. Thus, according to this we need at least two bonds to form an angle between them i.e. will require at least three atoms.
The angle depends upon the attraction and repulsion present in between the atoms due to the presence of electrons; and on the bond length.
For $N{{H}_{3}}$ and $P{{H}_{3}}$-
The size of the N atom is less than P and thus is more electronegative. Both have 4 pairs of electrons of which 3 are busy in bonding and one acts as a lone pair.
Due to the electronegativity, nitrogen pulls the electron pairs towards itself decreasing the bond length than the bond length of $P{{H}_{3}}$. As, electron comes closer they repel in the same space provided in $N{{H}_{3}}$ increasing its bond angle.
Numerically,
Bond angle of $N{{H}_{3}}$ = $107{}^\circ $
Bond angle of $P{{H}_{3}}$= $93.4{}^\circ $

Note:The bond length of $P{{H}_{3}}$ is comparatively smaller than that of $N{{H}_{3}}$ due to the larger size of P atom and due to increase in one p orbital. This is the main fact for the differences in bond angle between both the molecules which resemble some of the similar properties.