You open a terminal and type the sudo command.
I would encourage you to install synaptic. It is a very good apt GUI. It tells you about the programs and can update, upgrade, install, uninstall and fix some broken packages.
Open a terminal (System Tools> Mate Terminal>) and type <sudo apt-get update> then <sudo apt-get install synaptic>
Synaptic will appear under System>Adminstration>Synaptic Package Manager
Here is another post about the Broadcom drivers available. This shows the Broadcom drivers available in the Ubuntu Mate repository. Beyond that you may need to contact the manufacturer to get the correct driver. My card is an "Intel Corporation Centrino Advanced-N 6205[Taylor Peak] (rev34)" and the driver installed during Ubuntu Mate Installation. I have had previous Broadcom drivers and just installed all the software available and it worked, but I had two occasion were a kernel update did not work and I had to use the old kernel until the next upgrade for the Broadcom card to work again. I know it is not ideal, but you could try a linux supported WIFI dongle too.