Shouldn't probability density functions be in the form of
$$P(X\in dx) = \cdots$$
Why does the one for gamma distributions divide by $dt$?
$T_r =$ time of $r^\text{th}$ arrival after time $0$ in a poisson arrival process with rate $\lambda$.
And if I multiply both sides by $dt$, how am I supposed to calculate $dt$ on the right side?
No, $$\mathbb P(X\in dx) = f_X(x)\cdot dx$$
Therefore the probability that $r$-th arrival occures at interval $(t,t+dt)$ $$\tag{1}\label{1} \mathbb P(T_r\in dt)=f_{T_r}(t)\cdot dt. $$ By the other side, the event that $r$-th arrival occures at interval $(t,t+dt)$ means that exactly $r-1$ arrivals occure before $t$, and exactly one - on the interval $(t,t+dt)$. This events are independent, and the probability of first one is $$ \mathbb P(N_t=r-1)=\frac{(\lambda t)^{r-1}}{(r-1)!}e^{-\lambda t}, $$ while the second one has the probability $$ \mathbb P(N_{dt}=1)=\lambda \cdot dt. $$ Therefore $$\tag{2}\label{2} \mathbb P(T_r\in dt)= \frac{(\lambda t)^{r-1}}{(r-1)!}e^{-\lambda t} \cdot \lambda\cdot dt $$ Compare (\ref{1}) and (\ref{2}).