I believe you have misunderstood what the parsing table does. The parsing table tells for each pair $(A,a)$, where $A$ is a non-terminal symbol and $a$ is a terminal symbol, which grammar rule you have to use so that the next non-terminal symbol produced to the string from the left will be $a$. For example, the symbol $R_2$ produces either the symbol $id$ by the derivation
$$R_2 \rightarrow M) \rightarrow idR)$$
or the symbol $)$ by
$$R_2 \rightarrow )$$
Thus in the parsing table, $(R_2, id)$ points to rule "$R_2 \rightarrow M)$", and $(R_2,\textrm{'})\textrm{'})$ points to rule "$R_2 \rightarrow )$". The case $(R_2, \$)$ is a parsing error, since the dollar sign cannot be reached from $R_2$ without first producing either $id$ or $)$. If fact, all other terminal symbols except for $id$ and $)$ are parsing errors.
A grammar is a valid $LL(1)$ grammar if for every pair $(A,a)$ there is only one choice for which grammar rule to use. In our grammar the possibly problematic rules are the ones for $L_2$ and $R_2$, because they have two choices. The rest of the non-terminals are clearly unambiguous, as there is only one choice.
I gave the unambiguous parsing table rules $R_2$ above.
For $L_2$, the only symbols which do not produce a parsing error are $\epsilon$ and $::$. For $\epsilon$ we use the rule $L_2 \rightarrow \epsilon$, and for $::$ we use the rule $L_2 \rightarrow :: L$. Therefore the parsing rules for $L_2$ is also unambiguous, and in conclusion the grammar is a valid $LL(1)$ grammar.
