A narrowband signal seems like (almost) periodic as indicated by
$$ x[n] = m[n] \sin( w_0 n) $$ where the message $m[n]$ has such a low bandwidth that the peak amplitude (the envelope) of the carrier sine wave changes very slowly compared to how fast the sine wave oscillates between those +/- envelope limits. This makes its autocorrelation sequence also to be long tailed and almost periodic. And it means that the narrowband signal is highly correlated for long lags (long delays).
The opposite is true for the wideband signal, its correlation sequence ceases very rapidly even for a small amount of lag.
The Wiener filter works by estimating the supplied desired response signal from its filter input signal, whose success depends on the amount of correlation between the desired signal and the filter input.
If there is large correlation between the desired signal and the filter input, then the filter will successfully estimate the desired signal and the error (which is desired signal minus the filter ouyput) signal will be small. On the other hand if there is not enough correlation between the two, then the estimation will be poor.
Based on this operational principle of the Wiener filter, and based on the fact that the narowband and the wideband signals have different correlation characteristics, from the given adaptive filter scheme, you can explain its operation as follows.
First, you see that the filter input signal and the desired response signals are the same (except the delay) and are a mixture of an auto-correlated narrowband component plus an uncorrelated wideband component.
The delay is essentially used to decorrelate the desired response and the filter input. However, by carefuly chosing the amount of delay (the lag), you can insure that the wideband component is fully decorraleted whilst the narrowband component is still corelated enough. If you chose the delay to too long, both components will lose their cross-correlation to the desired response. If you chose the delay too short, you would not be able to suppress the wideband component enough. Some intermediate delay will be sufficiently decorrelating the wideband component while still maintaining the cross-correlation between the narrowband components at the desired signal and the filter input.
When this is the case, the filter will successfully predict the correlated narrowband component in the desired response from the delayed version of it at its input, and it will therefore be subtracted from the supplied desired response signal which yields as the narrowband component as the error signal.