1. Why $\inf U(f,P') \le U(f, P)$ and $\sup L(f, P') \ge L(f,P) $?
I tried to research but I can't find where Spivak defined it $P'$?2. Why are there two partitions P', P''? Not the same? P' is for the upper sum and P'' for the lower sum? I see the proof later defines $P = P' \cap P''$? Why not start with this one partition?
3. Intuition please on this result? I don't understand how an equality $\sup L(f,P) = \inf U(f,P)$ is equivalent to a strict inequality. I know the proof of theorem 1.2.6:
The book is just being sloppy there. $P'$ simply ranges over all possible partitions in the infimum and supremem. In other words, when it says $\inf \{ U(f,P') \}$, it means $$ \inf \left\{ U(f,P') \,|\, \text{$P'$ is a partition} \right\} \text{,} $$ and the same goes for the supremem.
Because $\inf \{ L(f,P) \} = \sup \{ L(f,P) \}$ doesn't guarantee (or at least not obviously guarantee) that there's a single paritition $P$ for which $$ U(f,P) - L(f,P) < \epsilon \text{.} $$ It only telly you that there is some paritition (say $P'$) for which $U(f,P')$ is closer to the supremem than $\epsilon/2$ (because if there weren't, it wouldn't be a supremem), and that simiarly there is some $P''$ for which $L(f,P'')$ is closer than $\epsilon/2$ to the infimum. Since the infimum and the supremem are the same, it follows that $U(f,P') - L(f,P'') < \epsilon$. Now that you have that, you can go looking for a single partition $P$ for which this holds. Which you do by looking for a $P$ which is finer than both $P'$ and $P''$, and which thus increses the lower bound and decreases the upper bound. Thus, for this single $P$, the relationship still holds, and voilá, you got your proof.
The result simpyl tells you that yes, integrable really just means that if you choose a fine enough partition, the lower bound and the upper bound are arbitrarily close together. In other words, both the lower and the upper bound converge to the same number as the partition gets finer and finer exactly if the function is integrable.