$$f_{T_r}(t)=\lim_{\Delta t\rightarrow0}\frac{P(t\le T_r\le t+\Delta t)}{\Delta t}$$
I get $\lim_{\Delta t\rightarrow0}P(t\le T_r\le t+\Delta t)$ because the purpose of this probability desity function is to find the probability near $t$, but why divide by $\Delta$t?
First, if we didn't divide by $\Delta t$ the we would get $0$ for the density: $$ \lim_{\Delta t\to 0}P(t\le T_r\le t+\Delta t)=0.$$
Second: By the definition of the density, if it exists then $$ P(t\le T_r\le t+\Delta t)=\int_t^{t+\Delta t}f_{T_r}(u)\ du\approx\Delta tf_{T_r}(t).$$
Dividing both sides by $\Delta t$ we get that $$ \frac{P(t\le T_r\le t+\Delta t)}{\Delta t}\approx f_{T_r}(t).$$
That is, if the limit in question exists for all $t$ then it has to equal the density.