Step 1: Recall what hyperconjugation needs.
Hyperconjugation is the overlap of a sigma bond with an adjacent empty $p$ orbital or a pi system. So an empty or partly empty orbital next door is essential.
Step 2: Look at the methyl carbanion.
In $CH_3^-$ the carbon is $sp^3$ hybridised and holds a lone pair. There is no empty $p$ orbital available.
Step 3: Judge Statement-I.
Because there is no empty orbital to accept electron density, $CH_3^-$ cannot gain hyperconjugative stability. So Statement-I is correct.
Step 4: Judge Statement-II.
Statement-II says there is no vacant $p$ orbital in $CH_3^-$ so it cannot take part in hyperconjugation. This is the true reason. So Statement-II is correct.
Step 5: Check the link.
The lack of a vacant $p$ orbital is exactly why hyperconjugation fails, so Statement-II correctly explains Statement-I.
Step 6: Choose the option.
Both statements are correct and II explains I.
\[ \boxed{\text{I correct, II correct and explains I}} \]