Generally the impact that it has occurs in a few ways.
Firstly a lot of the stuff you tend to want to remove in a sample is often located in the tail of the sample, including stuff like weird room resonances (if it was recorded) and any lingering frequencies that clash with other elements in your track. Fiddling with the ADSR values with a view to removing the tail of the sample, or reducing it, can often dramatically clean up a sample and make it sound better in context. If you find you want a little sustain back into the sample, adding some reverb can often help with this.
Next up, removing the tail of a sample can help if you use a lot of compression. Compression tends to bring up the sustain value of a sample, which may drastically alter the characteristics of the sample in question. You might choose a sample partly based on it's ADSR characteristics, then when you add compression you find that the processing impacts on these characteristics making the sample no longer fit the context of your track. Tweaking the ADSR values of the sample and dampening down the sustain portion prior to the audio going through the compression phase can in some instances help counter this.
Also, and this is quite a simple one, making sure the sample completely fades out before the next sample on the track plays can maximise the impact of the next sample. This works particularly well for transient material, like drum hits and percussive stuff. By the same token, if you are working with hi hats, cymbals and other such material, removing the initial attack portion of the sound to get it out the way of your main kick / snare sounds can make these sounds fit much better.