Given two random variables $X$ and $Y$ that are independent and exponentially distributed with the same parameter $\lambda$, and given that $Z = X + Y$, the joint density $f_{X,Z}(x,z)$ can be expressed as:
$$ f_{X,Z}(x,z) = f_X(x) f_Y(z - x) $$
Why is this true?
Imagine that the variables are discrete, like suppose that $X$ and $Y$ took values in the positive integers. Now consider the probability $$\mathbb{P}(X+Y = z).$$ This would be $$\mathbb{P}(X+Y = z) = \sum_{i=1}^{z-1} \mathbb{P}(X+Y = z | X = i) \mathbb{P}(X = i)$$ $$\sum_{i=1}^{z-1} \mathbb{P}(Y = z -i ) \mathbb{P}(X = i)$$ You can see that this is similar to the form you wrote above, where you have the $i$ and the $z-i$. We can rewrite the notation to be in line with yours; we can define $$f_X(m) = \mathbb{P}(X = m),$$ and we get $$f_{X+Y}(z) = \sum_{i=1}^{z-1} f_Y(z-i) f_X(i).$$
You can generalize this idea to continuous random variables, but instead of the summation, you have an integral sign.
