Why has Avidemux been removed from Ubuntu 16.04's repository?

I think nobody outside Canonical can answer that authoritatively, and an answer wouldn't be productive anyway.

If you want to know how to use recent avidemux versions on xenial, there are several PPAs around.

The ppa from Thanh Tung Nguyen seems to be of good quality, but unfortunately, the last release failed to build. There is also a getdeb.net ppa which specializes in obscure software, and it has the latest avidemux 2.6.12. You need to install the avidemux2.6-qt version or the avidemux2.6-cli version. For me, it runs without issues.

You can use the binaries from wily, they work under xenial as well (I use it frequently). Just get the .deb files from

https://launchpad.net/ubuntu/+source/avidemux/1:2.5.4-0ubuntu16

and install using

sudo dpkg -i FILENAME

If there are error messages you are still missing one of the deb files (read the error message and download the .deb file you need).

At the end do a

sudo apt-get install -f

to download the other dependencies automatically (libfaac, etc.)