how to find PDF of $X+Y$

Note that we are fixing a value of $t$; $s$ is a dummy integration variable so you cannot substitute it with anything. We have (due to independence) \begin{align} f_{X+Y}(t)&=(f*f_Y)(t)\\ &=\int_{-\infty}^{\infty}f(s)f_Y(t-s)\,ds\\ &=\int_{\{s\in\Bbb{R}\,:\, 0\leq t-s\leq 1\}}f(s)\cdot 1\,ds \end{align} The last equality is precisely for the reason you mentioned.

So, we're integrating over a certain set of $s$ values. This set depends on $t$, so you just have to rearrange the inequality "to make $s$ the subject of the inequality". For example, I'm sure you can verify that \begin{align} \{s\in\Bbb{R}\,:\, 0\leq 3-s\leq 1\}&= \{s\in\Bbb{R}\,:\, -1\leq s-3\leq 0\}\\ &=\{s\in\Bbb{R}\,:\, 2\leq s\leq 3\}\\ &=[2,3] \end{align} So, what is the result for a general value of $t$?