The aggregate claim is defined as $P(S \le x) = \sum_{i=0}^n P(N=n) P(S \le x | N=n) $.
The actual graph shows this instead:
of $f(s) = 0.1s$ and $f(s) = 0.2-0.01s$. Why? Did they integrate it? I understood that they used the formula for a convolution to solve it. Why does the graph look like that the actual formulas are $f(s) = 0.0025s + 0.025$ and $f(s) = -0.0025s + 0.05$ for $ 0 < s \le 10$ and $10 < s \le 20$ respectively.
Your formula gives the cumulative distribution function $$F_S(s) = \Pr[S \le s].$$ The question is asking for the frequency distribution, that is to say, the probability density $f(s)$ of $S$.
Therefore, you can certainly calculate $F_S(s)$ using your formula, but then you must differentiate it to get the graph of $f(s)$. Or you can use convolution to obtain $f(s)$ directly. It is important to note that the distribution of $S$ is partly continuous and partly discrete: there is a discrete mass at $S = 0$, as illustrated in the picture, corresponding to $\Pr[S = 0] = 1/2$. Then given that there is at least one claim, the size of the claim is a continuous random variable.