The dust limit is not actually fixed, technically - it varies based on the type of output. 546 satoshis is simply the most commonly known one, for a p2pkh output. Being the longest-lived output type, I suspect some wallets/blog posts/literature might treat it as a hard coded dust limit.
As to how to arrive at 546 satoshis, we must first know what "dust" means. A dust output is an output which costs more to spend, than it is worth. In other worsewords, an X BTC output that costs >X to spend, is a dust output. This is directly proportional to the amount of data required to spend an output, since fees in bitcoin are commonly denoted "per-byte". The more bytes you must add to your tx to spend an output, the higher its dust threshold.
A very basic tx consisting of 1 p2pkh input (~148 bytes), and 1 p2pkh output (~34 bytes) comes out to 182 bytes. The dust limit is 3 times this number (assuming a relay fee of 1 satoshi), or 182*3 = 546 sats.
For more complex txs, such as p2sh, this number is larger. For less space-intensive ones such as the newer segwit options, this number would be lower.
The code used in Bitcoin core to determine the dust threshold can be found here.
