This happens because of how Office documents are opened for editing.
When you open an Office document, the application creates a hidden temporary file in the same folder. As you work in the document, your modifications are written to this file (as an aside, this is how it's possible for Auto Save to work).
When you Save the document, the original document is deleted, then the temp file is renamed to that of the original. Because your users don't have permission to delete files, the operation fails.
Here's an overview of the process as detailed in this Microsoft Word support article:
Create temp file Create ~wrdxxxx.tmp
Write temp file Save example data to ~wrdxxxx.tmp
Delete original file Delete EXAMPLE.DOC
Move temp to target name Move ~wrdxxxx.tmp to Example.doc
According to the same article:
Word gains significant performance speed by placing the temporary file in the same directory as the saved file.....The location where Word creates the temporary files is hardcoded information and cannot be edited. Therefore, it is important that NTFS permissions for the user are set accordingly.
This is the same for across all Office desktop applications that save documents, i.e. Word, Excel, PowerPoint, and Publisher. A complete list of NTFS permission required for Word (specifically) can be found here. NTFS Delete is one of them.