Once you've found the sample or preset you are looking for you can simply download it by clicking the icon on the right-hand side of the sample or preset. You can also click Download All at the top right to batch download all samples (doesn't work for Presets currently). Lastly, you can shift select or batch select samples using the checkboxes on the left-hand side of the sample.
Samples are downloaded directly into the Splice app. Once they are downloaded you can search and filter them just like the website.
You can change the location of where the samples are downloaded, simply navigate to the Splice app's preferences and click Set New Folder. If you select an external drive make sure you QUIT SPLICE before ejecting the drive or else the app will go back to the default folder location.
Presets are not downloaded within the Splice app, they are downloaded directly to your browser's download folder. For more information on installing the presets once they have been downloaded please refer to this article.