The intuition is that, following any history $h^t$, the continuation strategy should be a subgame perfect equilibrium $\sigma|h^t$, which allows you to predict the value of continuation play ($V(\sigma|h^t)$) with certainty. Therefore, when choosing the current-stage strategy, a player know her action $a$ counts toward $h^t$, which in turn, leads to $\sigma|h^{t-1}\cup\{a\}$, generating a certain continuation payoff $v_i(\sigma|h^{t-1}\cup\{a\})$. And because other things are already determined, you can simply write $v_i(a)$. This is precisely why the continuation payoff is in fact history-strategy dependent.
The normal form says the players are choosing today's strategy $a$ by trading off today's payoff $u_i(a)$ and (discounted) future valuation $v_i(a)$. You can use the normal form because everything can be summarised into a current-stage variable $a$.