One nice way to see the two statements are equivalent is by using Venn Diagrams.
We can represent the sentence $\forall x (M(x) \rightarrow \neg D(x))$ in a Venn diagram as follows:
Explanation: When in a Venn diagram an area is shaded, that means that there do not exist any objects there. So, by shading the area that is both inside the '$M$' circle and inside the '$D$' circle, we are forcing that anything that is inside the '$M$' has to be outside the '$D$', and hence anything that is an '$M$' cannot be a '$D$' ... which is exactly what we want.
Now, compare this to the Venn Diagram for $\exists x (D(x) \land M(x))$:
This time, we put an 'X' in the intersection, indicating that there is at least one object there, capturing $\exists x (D(x) \land M(x))$
Now, if you compare the two Venn Diagrams, you'll notice that they are saying the exact opposite: one is saying that there is not a thing in the intersection, while the other is saying that is is a thing in the intersection. Thus, one is the negation of the other. And thus:
$\forall x (M(x) \rightarrow \neg D(x)) \Leftrightarrow \neg \exists x (D(x) \land M(x))$