I am assuming the cubes have faces numbered 1-6, since it wasn't specified.
Let us look at the first question: Why are the events "The blue cube shows a 4" and "The product of both cubes is less than 20" dependent?
Let's call A the event "The blue cube shows a 4" and B the event "The product of both cubes is less than 20." For two events to be independent means that the probability of the first happening is the same whether or not we know that the second has happened, and vice versa. That is, if A and B were independent, we would have
P(A) = P(A | B) and P(B) = P(B | A) where P(B | A) is the probability of event B given that we know A has occured.
Assuming the cubes are fair, we have that P(A) = 1/6. Remember that A represents the blue cube rolling a 4. Note that if we assume that A is true, then the possible products of both dice are given by
Blue = 4
Yellow: 1 2 3 4 5 6
Product: 4 8 12 16 20 24
Each of these outcomes is equally likely, so there is an exactly 4/6 possibility that the product is less than 20 given that the blue cube shows a 4. Therefore,
P(B | A) = 4/6 = 2/3
However, if we consider all possible rolls of two cubes, we see
Blue \ Yellow: 1 2 3 4 5 6
1 | 1 2 3 4 5 6
2 | 2 4 6 8 10 12
3 | 3 6 9 12 15 18
4 | 4 8 12 16 20 24
5 | 5 10 15 20 25 30
6 | 6 12 18 24 30 36
Of these 36 outcomes, exactly 28 have product less than 20. Since each outcome is equally likely, the probability that the product of the two cubes is given by
P(B) = 28/36 = 14/18 = 7/9 > 2/3 = P(B | A)
So it is less likely for the product of two cubes to be less than 20 if we know that one of the cubes reads a 4. Thus the events are dependent. If they were independent, we would have found the same probability for both P(B) and P(B|A).
Now that we have laid the groundwork, question 2, find P(A and B) is easy. In general, we have
P(A and B) = P(A) * P(B | A)
We know that P(A) = 1/6 since the cube is fair, and P(B | A) was found to be 2/3. Therefore
P(A and B) = 1/6 * 2/3 = 1/9
