Why are the files downloaded while Installing MS Office fonts "exe"?

Because the exe's contain the fonts - I don't think they are installed as Window does them, they are extracted (EDIT: on windows they self-extract, thanks @JakeGould)

To see this, you can download one (I did this one), and open it up in file-roller (Archive Manager):

Ubuntu and other people probably can't distribute the fonts directly to licensing things from MS - the exe's probably count as 're-distributable' under the EULA linked in the installer's prompt:

Including this bit and probably other stuff in it:

1. GRANT OF LICENSE. This EULA grants you the following rights:

• Installation and Use. You may install and use an unlimited number of copies of the SOFTWARE PRODUCT.
• Reproduction and Distribution. You may reproduce and distribute an unlimited number of copies of the SOFTWARE PRODUCT; provided that each copy shall be a true and complete copy, including all copyright and trademark notices, and shall be accompanied by a copy of this EULA. Copies of the SOFTWARE PRODUCT may not be distributed for profit either on a standalone basis or included as part of your own product.

2. DESCRIPTION OF OTHER RIGHTS AND LIMITATIONS.

• Limitations on Reverse Engineering, Decompilation, and Disassembly. You may not reverse engineer, decompile, or disassemble the SOFTWARE PRODUCT, except and only to the extent that such activity is expressly permitted by applicable law notwithstanding this limitation.
• Restrictions on Alteration. You may not rename, edit or create any derivative works from the SOFTWARE PRODUCT, other than subsetting when embedding them in documents.

MS Office fonts are fonts from Microsoft. Microsoft doesn't make binaries for other OS like Ubuntu, thus ttf-mscorefonts-installer download various MS fonts in exe format and then cabextract extracts fonts from these exe files.

The MS Core Fonts come in a self-extracting zip archive we can open or install without the need to run the Windows executable or the included Windows font installer program.

To install a font simply open the font.exe file with the Archive Manager to view the contents of this archive.

We then install the contained TTF font with a double click on the corresponding file to open it with the Font Viewer. There we will have an Install button on the top right which will install the font to ~/.local/share/fonts without any further questions asked:

After downloading, the fonts you need will be extracted from these exe files. You can see the extracting progress from the output in the console. There is no answer contains the output, so I post it here (please note the package "cabextract" and the content of the license in the middle):

$ sudo apt install ttf-mscorefonts-installer
...
The following NEW packages will be installed:
  cabextract libmspack0 ttf-mscorefonts-installer
...
ttf-mscorefonts-installer: processing...
ttf-mscorefonts-installer: downloading http://downloads.sourceforge.net/corefonts/andale32.exe
...

These fonts were provided by Microsoft "in the interest of cross-
platform compatibility".  This is no longer the case, but they are
still available from third parties.

You are free to download these fonts and use them for your own use,
but you may not redistribute them in modified form, including changes
to the file name or packaging format.

Extracting cabinet: /var/lib/update-notifier/package-data-downloads/partial/andale32.exe
  extracting fontinst.inf
  extracting andale.inf
  extracting fontinst.exe
  extracting AndaleMo.TTF
  extracting ADVPACK.DLL
  extracting W95INF32.DLL
  extracting W95INF16.DLL

All done, no errors.
...

All fonts downloaded and installed.
Processing triggers for fontconfig (2.13.1-2ubuntu3) ...

Since "you may not redistribute them in modified form, including changes to the file name or packaging format", package maintainers have to keep them as they are, even though these files are CAB archives ("cabinet" is CAB).