Q:

How does apt install work?

How does 
apt install chromium

install chromium. All I understand about apt is that it downloads the .deb files and then runs dpkg. Is there anything else apt does?

A:

Just simple answer is :
apt can be consider as front-end for dpkg  
But more than that apt is complete package manager for update,install,upgrade,dist-upgrade and a lot of other things check this  https://help.ubuntu.com/community/AptGet/Howto 
sudo apt update checks all the configured source for latest package available there and it downloads all the details for package available for installation.you can know a lot by man apt
update (apt-get(8))
           update is used to download package information from all configured
           sources. Other commands operate on this data to e.g. perform
           package upgrades or search in and display details about all
           packages available for installation.

Now on performing sudo apt install <package> if there is package available for installation it start downloading the package (.deb file) and perform dpkg to install the package. Otherwise if there is no package availabe it throw error message.

