Base Substitution Probabilities Derivation
(1) There are two cases. For the first equation (6), the case $i \neq j,$

$$(1) \hspace{13mm} P_{ij}(dt) = u~dt ~\pi_j $$ if we have a transition from one base to another (for example G to C). In the case $i = j,$

$$(2)\hspace{10mm} P_{jj}(dt) = (1 - u~dt) + u~dt~\pi_j $$ holds for a same-base transition (G to G) or a non-transition (say, G and no change). Intuitively, we expect that, all things being equal, the probability of a base to same-base situation is slightly higher: it is the sum of probability that the base changes to itself plus the probability the base sits and does nothing in time $dt$ (with the same net result). 



(2) Now we look at the equation(s) (7). Again there are two cases.

$$(1a) \hspace{10mm} P_{ij}(t) = (1 - e^{-ut})\pi_j $$

For case 2 (base to same base transition, $i = j$): 

$$(2a)\hspace{13mm} P_{jj}(t) = e^{-ut} + (1 - e^{-ut})\pi_j. $$



(3) The author uses  $P(dt),$ $P$ at a very small (infinitesimal) quantity $dt.$ He tells us the leap from (1) to (1a) follows "almost immediately" when we see that $e^{-ut}$ is the probability that no base transition occurs. Well, then $(1 - e^{-ut})$ is the probability that a transition occurs.

Now $e^{-ut} \approx (1 - ut) $ (a truncated Taylor series) or $ut\approx (1 - e^{ut}).$ So for short times $dt$ we have $u~dt \approx (1 - e^{-u dt})$ and this explains his first equality $P(dt) = u~dt~\pi_j.$ 

So we have to recognize when he tells us about $e^{-ut}$ that $u t$ is the [truncated] Taylor series representation of $(1-e^{-ut})$ for $t$ very near 0. That is how he gets from 1 to 1a and (6) to (7). 



(4) $(1 - u)$ is the probability that no transition from one base to another occurs in some short time $t_i.$ The probability that this happens over n consecutive time intervals is $(1 - u)^{(nt_i)}.$ Let $nt_i = T.$

Note that  $u \ll 1.$ We can write $(1 - u)^T = (1 - \frac{uT}{T})^T $ and 
$$\lim_{T \to \infty} (1 - \frac{uT}{T})^T= e^{-uT}.  $$ This reasoning comes directly from that used to derive the Poisson distribution which is used to study rare events. See, for example, Deonier, Computational Genome Analysis, p.74.