Step 1: Tie acidity to how comfortably the resulting anion sits.
When a phenol loses its acidic proton, it forms a phenoxide ion. The more that negative charge can be spread out and calmed down, the easier it was to remove the proton in the first place, so acidity really comes down to how stable that phenoxide ion is.
Step 2: See what the nitro group does to that anion.
The nitro group is strongly electron withdrawing, both by pulling density through the sigma framework and by resonance, so it happily accepts and spreads out the extra negative charge sitting on the phenoxide oxygen. This makes o-nitrophenoxide considerably stabilised, so o-nitrophenol gives up its proton easily and is fairly acidic.
Step 3: See what the methoxy group does instead.
Methoxy is an electron donating group through resonance, it pushes its own electron density into the ring, which actually piles more negative charge onto an already negative phenoxide oxygen. That clash of like charges destabilises o-methoxyphenoxide, so o-methoxyphenol resists losing its proton and is markedly less acidic.
Step 4: Compare and conclude.
Because the nitro group stabilises the conjugate base while the methoxy group destabilises it, o-nitrophenol turns out to be considerably more acidic than o-methoxyphenol.
\[ \boxed{\text{o-Nitrophenol is more acidic since } -NO_2 \text{ stabilises the phenoxide ion, while } -OCH_3 \text{ destabilises it}} \]