Subtraction of a negative number
Two ways to explain this.
One way to understand what subtraction is: $a-b = a+(-b)$. When you subtract $b$ from $a$, you are actually adding $-b$ to $a$. This is the way $a-b$ is defined. With this definition, you know that $a-(-b)$ equals $a+ (-(-b))$. Now, what is the number $-(-b)$? Well, by definition, for any $c$, $-c$ is the number that satisfies the equation $c+(-c) = 0$. This means that the number $x=-(-b)$ is the number that solves the equation $(-b) + x = 0$. Of course, it is obvious that, since $(-b) + b = 0$, this means that $-(-b) = b$, and that means that $$a-(-b) = a+(-(-b)) = a+b.$$
In the real line, you can see the number $a+b$ as the number you get when you add the lines $a$ (line from $0$ to $a$) and the line from $0$ to $b$ and put them one after the other (end to end). Subtraction in this sense means that calculating $a-b$ is taking the line from $0$ to $a$ and the line from $b$ to $0$ and putting them end to end. But since the line from $-b$ to $0$ is the same as the line from $0$ to $b$ (just shifted), this also means subtracting $-b$ is the same as adding $b$.
To calculate $a+b$, you take the line from $0$ to $a$ (directed to the right) and the line from $0$ to $b$ and but the end of one to the beginning of the other, so you have something that looks like
:------>:------------>
a b
:-------------------->
a+b
However, if you are subtracting $b$, you must reverse the direction of the line which belongs to the number you are subtracting, so you have
:------------------>
a <------:
b
:----------->
a-b
So, the point is that to add $b$ to $a$, you add the line from $0$ to $b$ to the line from $0$ to $a$. If you are subtracting it, you have to flip the direction of the line you are subtracting. The main idea is that if you flip the direction of $b$ and add it to $a$, it's the same thing as if you took the line for $-b$.
$$\begin{align} a - (-b) &= a +\color{red}{0} - (-b) \\ &= a + \color{red}{\left(\; b + (- b) \;\right)} - (-b) \\ &= a + b + \color{blue}{\left(\;(-b)-(-b)\;\right)} \\ &= a + b + \color{blue}{0} \\ &= a + b \end{align}$$
The idea is this: We "can't" subtract $(-b)$ from an expression ---in this case, $a$--- that doesn't have a $(-b)$ in it, so we cleverly manipulate the expression into an equivalent form that does have a $(-b)$ in it. The easiest way to do this is to add zero (which changes the expression's value not at all), and immediately interpret that as the sum of $b$ and its negator, $(-b)$. At that point, we have a $(-b)$ in our expression, from which we can subtract $(-b)$, achieving our goal. We find ourself with another $0$, which we can ignore.
I'll note that a similar strategy works to explain the rule for "dividing by a reciprocal": $$\begin{align} a \div \frac{1}{b} &= a \cdot \color{red}{1} \div \frac{1}{b} \\ &= a \cdot \color{red}{\left(b \cdot \frac{1}{b}\right)} \div \frac{1}{b} \\ &= a \cdot b \cdot \color{blue}{\left(\frac{1}{b}\div\frac{1}{b}\right)} \\ &= a\cdot b \cdot \color{blue}{1} \\ &= a b \end{align}$$