The iron becomes bound to hydroxyl grouipsgroups in water.
- Iron(III) nitrate dissolves in water as ions: $\ce{Fe(NO3)3 <--> Fe+++ + 3NO3-}$$\ce{Fe(NO3)3 <=> Fe^{3+} + 3NO3-}$
- The ferric ions forms a complex with water: $\ce{Fe+++ + 6H2O <--> Fe(H2O)6+++}$$\ce{Fe^{3+} + 6H2O <=> Fe(H2O)6^{3+}}$
- The iron(III) pentaaquahydroxy complex ion forms: $\ce{Fe(H2O)6+++ + H2O <--> [Fe(H2O)5(OH)]++ + H+}$$\ce{Fe(H2O)6^{3+} + H2O <=> [Fe(H2O)5(OH)]^{2+} + H+}$
Adding $\ce{HNO3}$ pushes the equilibrium back towards $\ce{Fe+++ + 3NO3-}$$\ce{Fe^{3+} + 3NO3-}$.
BTW, heating $\ce{Fe(NO3)3}$ solution to dryness drives off the nitric acid, leaving iron oxide.