Horse's downloads list lives at the bottom of the sidebar, not in a separate window or tab. Files themselves still land in your usual download folder.
Viewing your downloads
Files appear at the bottom of the sidebar as they download. X removes a single entry from the list. The folder icon at the top of the list opens the download folder on your desktop. The trash icon clears the list.
Where files actually live
Your default download folder, set by the operating system. Clearing the sidebar list does not delete the files, it just hides the entries.
Showing and hiding the list
Mouse users: Menu > View > Show Downloads (or Hide Downloads)
Keyboard users: ⌘+⇧+J: same shortcut toggles it back off.