Step 1: Understand electron gain enthalpy.
Electron gain enthalpy is the energy change when a gaseous atom accepts an electron. A more negative value means the atom releases more energy and so has a stronger tendency to gain an electron. We compare $F$, $O$, $S$ and $N$.
Step 2: General period and group trend.
Across a period electron gain enthalpy generally becomes more negative, and down a group it usually becomes less negative because the added electron sits farther from the nucleus.
Step 3: Place fluorine.
Fluorine is the most electronegative element with a high effective nuclear charge, so it has a very high tendency to gain an electron and stands at the top of the order.
Step 4: Compare oxygen and sulphur.
Although oxygen lies above sulphur in group 16, oxygen is a very small atom. The incoming electron faces strong electron electron repulsion in the compact $2p$ shell. Sulphur, being larger, accommodates the electron more comfortably, so sulphur has a more negative electron gain enthalpy than oxygen. Hence $S \gt O$.
Step 5: Place nitrogen lowest.
Nitrogen has a stable half filled $2p^3$ configuration, so adding an electron is unfavourable and its electron gain enthalpy is the least negative among these. So nitrogen comes last.
Step 6: Assemble the order.
Combining all comparisons we get \[ F \gt S \gt O \gt N \] which matches the key.
\[ \boxed{F \gt S \gt O \gt N} \]