What is the concrete mathematical argument why $a^3+a=b^3+b\Rightarrow a=b$ is true in $\mathbb{R}$? The question arose from the following proof:
$f(x)=x^3+x+1$ is injective:
Let $a,b\in \mathbb{R}$ and $f(a)=f(b)$:
$f(a)=f(b)\Leftrightarrow a^3+a+1=b^3+b+1\Leftrightarrow a^3+a=b^3+b \Rightarrow a=b$
Is $a^3+a=b^3+b\Rightarrow a=b$ mathematically rigorous enough? I don't know the concrete mathematical argument why $a^3+a=b^3+b\Rightarrow a=b$ is true?

What do you mean by "rigorous enough"? No argument is in your step $a^3 + a = b^3 + b \Longrightarrow a = b$.  Some reasoning should be included there.
Method 1. Use calculus to show $x^3 + x$ is an increasing function.  That it should be an increasing function is obvious by staring at a graph of it, and calculus allows you to explain why the function is increasing.
Method 2. Use algebra to show $x^3 + x$ is an increasing function.  If $x < y$, then we want to show $x^3 + x < y^3 + y$. Write $y = x + h$ for $h > 0$. Then
$$
y^3 + y = (x+h)^3 + (x+h) = x^3 + 3x^2h + 3xh^2 + h^3 + x+h,
$$
so
$$
(y^3 + y) - (x^3 + x) = 3x^2h + 3xh^2 + h^3 + h.
$$
We want to show the right side is positive for all $h > 0$, no matter what $x$ is. View the right side as a quadratic polynomial in $x$:
$$
(3h)x^2 + (3h^2)x + (h^3 + h).
$$
Its discriminant is $(3h^2)^2 - 4(3h)(h^3 + h) = -3h^4 - 12h^2 < 0$, so that quadratic polynomial in $x$ has no real roots: its values as a $x$ varies are either always positive or always negative.  At $x = 0$ its value is $h^3 + h$, which is positive, so its values at all real $x$ are positive.
Method 3. Use algebra to show that if $a^3 + a = b^3 + b$ then $a = b$. Rewrite the equality as $a^3 - b^3 = b - a$. Factoring the left side,
$$
(a-b)(a^2 + ab + b^2) = b-a = -(a-b).
$$
If $a \not= b$ then the term $a-b$ is nonzero, so you can divide by it to get
$$
a^2 + ab + b^2 = -1.
$$
If $a$ or $b$ is $0$ then this equation becomes $b^2 = -1$ or $a^2 = -1$, which have no solution (in real numbers). If $a$ and $b$ are both nonzero and have the same sign, then all three terms on the left side are positive, so their sum is not $-1$. If $a$ and $b$ are both nonzero with opposite signs, then rewrite the left side:
$$
(a+b)^2 - ab = -1.
$$
Here $(a+b)^2 \geq 0$ and $-ab > 0$ (since $ab$ is negative due to the opposite signs), so again such an equation is impossible.
