What is the significance of a spike like autocorrelation function
Consider a linear time-invariant system with unknown impulse response $h(t)$. If the input is a zero-mean wide-sense-stationary random process $\{X(t)\}$, then, with $\{Y(t)\}$ denoting the output process, we have that
$$R_{Y,X}(\tau) = E[Y(t)X(t+\tau)] = h\star R_x\big|_\tau.$$
So. if $\{X(t)\}$ is a white noise process wth autocorrelation function
$K\delta(t)$, then $R_{Y,X}(\tau) = Kh(t)$. Why is this at all important? Well, most processes are assumed to be ergodic, and so $R_{Y,X}(\tau)$ can be estimated by cross-correlating a sample path of the output with a sample path of the input with a delay of $\tau$. Thus, the spikiness of the autocorrelation function has a role in system identification of unknown systems.