If I have a trading strategy such that at each time $t$ I own $\Delta_t$ units of stock $S_t$ and $\psi_t$ units of bond $B_t$, it is a replicating strategy for some claim with time $T \geq t$ payoff $X$ if the value of this portfolio $V_T = X$.
Written out, the time $t$ value of the portfolio is $V_t = \Delta_t S_t + \psi_t B_t$, and is self-financing if $dV_t = \Delta_t dS_t + \psi_t dB_t$; that is, changes in value are brought on only by changes in asset prices, not the strategy.
Now, why is a self-financing strategy so important for option pricing? If $V_T = X$ but this strategy is NOT self-financing, doesn't the time $t$ price of the option still have to be $V_t$, else arbitrage? If it's NOT self-financing, there may be cash injections/consumption along the way to the payoff, but so be it - it still replicates the payoff, so at each $t$ the price of the option must still be $V_t$.
Please let me know how I've confused myself so badly!