Once you've found the Sounds you are looking for, you will first need to add them to your library by clicking the + button to the right of the entry (the + button will appear once you hover your mouse over the selection).
You can click Get Pack at the top right of a sample pack to batch download all the samples in the pack. This does not include presets, MIDI, or any pack extras.
You can also shift-select multiple samples using the checkboxes on the left-hand side of the samples to batch-download them.
If you need to sync a Sound locally to your Splice folder, you can do so by clicking the cloud download button to the right of a Sound in the desktop app. You can also use the Sync Sounds feature in your app preferences to sync all your Sounds locally to your Splice folder. To get to your preferences, click on the icon at the top right of the desktop app.
When you download samples, they will appear in the Splice desktop app under the 'Sounds' tab. From there you can either download them onto your hard drive, click and drag them into your DAW, or copy them to your clipboard to paste in a location of your choice. The samples are automatically downloaded to your Splice folder:
Mac: Macintosh HD/Users/[your username]/Splice