Find W/L and Vt of NMOS
The ratio in currents is useful here because current is proportional to the square of overdrive voltage (called \$V_{gt}\$ here and \$V_{ov}\$ in some other courses)1. By taking the ratio in currents, you easily get the ratio of overdrive voltages, which is useful in some problems such as this one.

The notation appears to be a result of computing \$V_{gt}\$ at the two provided operating points which were implicitly numbered 1 and 2. \$V_{gt1} = 0.9\,\text{V} - V_t\$ and \$V_{gt1} = 0.8\,\text{V} - V_t\$.

Given \$V_{gt1}/V_{gt2} = 1.5\$, you know that \$\frac{0.9\,\text{V}-V_t}{0.8\,\text{V}-V_t} = 1.5\$. This can be easily seen to be satisfied by \$V_t = 0.6\,\text{V}\$ since 0.3/0.2 = 1.5.

1 As a simplified model that holds well for strongly-inverted FETs.