These really are based on the definitions of multiplication/division.
When you divide two numbers, you are simply asking "how many groups of x fit into y". Take for example, 6 divided by 2. Ask yourself "how many groups of 2s fit into 6?" Well, we know the answer is 3 groups of 2.
Now consider the case you mention: dividing by .5 (aka one-half or 1/2). If you divide 6 by 1/2 (or 6 by 0.5), just ask yourself, how many 1/2s fit into six?
two 1/2s would fit into one
four 1/2s would fit into two
six 1/2s would fit into three
eight 1/2s would fit into four
ten 1/2s would fit into five
twelve 1/2s would fit into six.
So, 12 groups of 1/2s would fit into six. This makes sense since for each of the 6 pieces, you are cutting them in half.
The case for multiplication is similar. Ask yourself "what do I get when I form together x groups of y?" If x is 3 and y is 2, then you'll end up with 2 groups of 3, which is 6.
Likewise, when multiplying by .5 (aka one-half) ask yourself, "what do I get if I form together a half-group of something?" then you'll notice that the number gets smaller. What do you get when you form together half of a group of 6? The answer is only 3. In fact, you get two groups of 3, each of which is half of 6.