What is the value of ab?
(1) |a| + b = a
(2) a + |b| = b

A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not.
B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not.
C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient.
D. Either statement BY ITSELF is sufficient to answer the question.
E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.

(C) Statement (1) is |a| + b = a. If 0 > a then |a| = -a. So the equation transforms into -a + b = a. This yields b = 2a. Therefore ab = 2a². But we do NOT know the value of a. So statement (1) by itself is NOT sufficient.

Statement (2) is a + |b| = b. Similarly to the previous reasoning, if 0 > b then |b| = -b. So the equation transforms into a – b = b. This yields a = 2b. Therefore ab = 2b². But we do NOT know the value of b. So statement (2) by itself is NOT sufficient.
If we use the both statements together, we can add the equations.
|a| + b + a + |b| = a + b
|a| + |b| = 0
Any absolute value is non-negative. So the sum equals 0 if and only if each absolute value is 0. Therefore a = 0, b = 0 and ab = 0. Statements (1) and (2) taken together are sufficient to answer the question, even though neither statement by itself is sufficient. The correct answer is C.
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I did not understand the concept of |a| = -a.

Take a look at the numerical example:
|4| = 4
|-4| = -(-4) = 4

Now in general, if a is a positive number, then
|a| = a

if a is a negative number, then
|a| = -a, because (-a) is positive.

a and b are variables.

Any absolute value is non-negative. If a and b were not zeroes, then |a| would be positive and |b| would be positive.
positive + positive > 0
That would contradict the equation. Even if only one variable was not zero, the left side would still be positive. So the both variables can possess only one value, 0.

If to plug in a = -b we will get |-b| + |b| as the left side of the equation. It is not necessarily 0. Plug any non-zero value for b, for example b = 2 yields |-2| + |2| = 2 + 2 = 4. So the conclusion "It could mean a = -b" is not correct.

If to solve the equation |-b| + |b| = 0:
The absolute values |-b| and |b| are the same (equal), so the equation transforms into
2|b| = 0
b = 0

No, it does NOT. For example, take a = -3 and b = -6:
|-3| – 6 = -3

The same is with the reasoning for Statement (2).