What
A checksum is a small value calculated from a larger block of data. It's used to quickly check if that data has been accidentally altered or corrupted. If the checksum of the data now matches the checksum of the data then, you can be reasonably sure the data hasn't changed. It's a simple integrity check.
Think of it like this simple analogy. You have a shopping list with prices:
Apples: $2
Milk: $3
Bread: $1
Total (Checksum): $6
You tell your friend the list and the total ($6). If your friend writes down:
Apples: $2
Milk: $5 (Oops, a typo!)
Bread: $1
New Total (Checksum): $8
They compare their new total ($8) with your original total ($6). Since they don't match, they know their list isn't right, even if they don't know which item is wrong.
Why
Just like the above example situation, you would want to verify the information you have. This confirms that the file was properly downloaded and that your local file is an exact copy of the file you wanted to download off from the server. Therefore, you may assume that the file wasn't corrupted or tampered.
What situation may call for it? On February of 2016, there were attacks on the Linux Mint website. The hackers modified the Linux Mint ISO, with a backdoor, and managed to hack the website to point to said file. Based on available information, they haven't modified the checksums available in the download page... so if you may have verified the ISO file, you would easily find out that there's something wrong with or during transfer.
How
It's pretty simple. Linux Mint provides relevant instructions regarding this (see attached image). You would need to download the sha256sum.txt and sha256sum.txt.gpg. After downloading the ISO file, cd to its containing directory and run sha256sum -b yourfile.iso. You would need to compare the command output to the unaltered contents of sha256sum.txt. If the sums match, your ISO image was successfully downloaded. If they don’t, download it again.