Can anyone explain how line 3 follows from line 2? IDK how Fubini's works with improper integrals. I just know that $\displaystyle\int_a^b\int_c^d g(x,y)\,dx,\,dy = \int_c^d\int_a^b g(x,y)\,dy\,dx$ holds for non-neg functions (which the PDF below is) but IDK how that allows the inner integral to start at $t$.
This is just a matter of recognising that $$ \{(x,t): x\ge 0 \wedge 0 \leq t \leq x\}=\{(x,t): t \ge 0 \wedge x \ge t\} $$
Just sketch the two sets and you'll see... This is why you have that $$ \int_0^{+\infty} \int_0^x g(x,t) \, dt dx = \int_0^{+\infty} \int_t^{+\infty} g(x,t) \, dx dt. $$