Comparing DFA's produced languages
L(M) ⊂ L(M') can be proved as below:

Suppose a string 'w' ∈ L(M') .Since it is given that F is a proper subset of F' , there exists a state q1 such that 
q1 ∈ F' but not to F.
Since the transition function is same for both the DFAs, hence there is a transition possible as below:
δ(q0, w) = q1which means the state change takes place from the initial state to the state q1 on reading the input symbols from the string 'w'.

Since q1 ∈ F' but not to F, we can say that 'w' does not belong to L(M).
On the same lines it can also be proved that any string in L(M) is also present in L(M'), thereby proving the relation.