Why we measure carbon?
How much hotter our world gets depends on the concentration of greenhouse gas emissions in the atmosphere, and on how potent those greenhouse gases are.
The concentration of these gasses depends on how much greenhouse gas we emit now, and how much we have emitted historically. When we measure these emissions, we want to measure how much we emit, and how potent each particular gas is. Because carbon dioxide is the greenhouse gas that is causing the most harm/warming, we focus on that, we then convert the impact of any other gases into a carbon dioxide equivalent (CO2eq) based on how potent they are. That potency is referred to as 'Global Warming Potential' (GWP) with Carbon Dioxide CO2 having a GWP of 1.
If a gas has a GWP of 20 it means that a tonne of that gas has a warming effect 20 times that of CO2, so it would be 20 kgCO2eq.
Why we use mass
Because carbon dioxide is a gas and floats around in the air, we tend to think of it as a volume rather than a mass. The problem with that is that the same amount of carbon can have different volumes at different pressures, so it's not a very consistent way of measuring things. Another option would be to measure the number of carbon dioxide molecules we emit - but that would be pretty impractical. Instead we choose to measure the mass of carbon dioxide emitted. It's much easier to do than counting molecules and, because every carbon dioxide molecule has the same mass, it's a consistent measure of the amount of carbon dioxide.
There are lots of units of mass. We could choose kg, stones, pounds, grams, mega tonnes etc. We use metric units like kilograms and tonnes because they are scientific units which are easy to work with. Out of those, we choose tonnes because they happen to be the right scale for our personal portion of emissions. In the UK, the average household has a carbon footprint of about 8 tonnes. About 4 tonnes of that is from electricity and heating, the rest is from transport, aviation, and waste. Feel free to call that 0.08 Megatons, 8,000kg, 8,000,000g, or even 282,192 ounces if you prefer, but we like to keep life simple and call it 8 tonnes.