What is the meaning of $Ta_k$ of fourier series or transform?
The idea is that a Fourier series is only defined for periodic signals. In the discussion in the linked slides, the author is considering a rectangular pulse train with period $T$. That is, a pulse of width $2S$ repeats periodically with a spacing of $T$ between them. The pulses are therefore centered at:

$$
[\ldots, -2T, T, 0, T, 2T, \ldots ]
$$ 

Now, consider what happens as $T \to \infty$: in the limit, the only pulse that remains is the one centered at zero; the others are infinitely far away. When the author makes the claim that:

$$
\lim_{T\to \infty} T a_k = E(\omega)
$$

He or she is trying to show that the Fourier transform, which is defined for suitably well-formed aperiodic signals, can be thought of as the Fourier series of that signal (which typically wouldn't be defined since the signal is not periodic) in the limiting case of an infinite period. Stated a little differently, you can in some way think of an aperiodic signal as a periodic signal with infinite period.

The multiplication by $T$ in the limit is to account for the differences in definition between the Fourier series and Fourier transform: the series representation typically has a factor of $\frac{1}{T}$, while the transform does not. I don't know that there is a lot of insight to be gained via this analysis, but it shows that the series and transform representations are intimately related.