Look at the resolution and file size for a given video clip as shot, say maybe something 5 minutes long or so. Now look at the same file after it has been transferred to your phone and compare its resolution and file size. This will tell you if the software is compressing the file to make it easier to copy to your phone and more suitable for display on your phone's screen. The smaller compressed file should still look pretty good on your phone though its quality will likely suffer if the same file were played on your monitor or an HD TV.
Compared to still image files, video files are humongous. I don't think a lot wireless hardware or software is really designed to handle these huge files as efficiently as a batch of smaller still image files. So to avoid the bottlenecks and slowdowns the video files get reduced in quality and file size, or with some devices they may not even attempt to transfer them at all.
