I think it's easier to answer this question qualitatively if you turn it around: why does a small range of wavelengths produce a broad wave packet, and vice versa?
Set up the waves so that they're all in phase at one point (say, the origin). If their wavelengths are nearly the same, they are still nearly in phase and interfere mostly constructively at a large distance from the origin. On the other hand, if they have a wide range of wavelengths, the destructive interference is significant at a short distance from the origin.
[itex]A(k)[/itex] gives the amplitude of the wave with wavelength [itex]2 \pi / k[/itex]. Although we normally write the integral using in infinite range (limits) in [itex]k[/itex], what counts is the range where [itex]k[/itex] is significantly different from zero.
You might say that a large range in [itex]k[/itex] has a larger number of waves in it (i.e. a larger number of values of [itex]k[/itex]), but that's not really correct because any continuous range has an infinite number of values in it!
In reality, most physicists would say that fourier transforms are simple.
However try building a wave packet graphically -- take several waves with different frequencies, and add them, by hand or whatever, and graph the result. Add some more -- big freqs and small ones. As you do this, you will see the interference patterns that ultimately produce a delta function -- make sure that all waves are of the form exp(kx), where k is the wave number, so that the exp functions are all = one at x=0.
It's not easy to explain it from scratch, and let alone, to understand it. But if you really want to know, pick up "Vibrations and Waves in Physics" by Iain G. Main (). It is explained as simply as it can be.