Economic Theory:
To understand an NFT, and their value one must first understand cryptocurrency itself from a value perspective. Most people assume crypto's usability as an unstoppable payment system (made famous by the dark web) is the foundation of it's "intrinsic" value. They're wrong. It's core value is, and always has been collectibility from the moment cryptocurrency was invented, proceeding it's intended use as a currency.
Because it can't be counterfeited by definition, Bitcoin's very first role was as a "digital collectible" first and foremost. The first major exchange for it was a service created for gamers to trade items they won in role playing games, and was named Magic The Gathering Online Exchange AKA Mt. Gox.
In almost the exact manner that Gold was first valued as a collectible, before it was used as a currency, this was also bitcoin's first use case. Drug dealers would have never been able to start using it, if bitcoin's collectibility wasn't providing it's initial "liquidity" AKA a reliable market to sell it by it's desirability as a collectible.
This often overlooked use case for cryptocurrency is key to understanding NFT's, and why they are potentially so valuable.
Technical Perspective:
NFT stands for non-fungible token.
First, it helps to understand exactly what a token is. In this context, a token is a digital asset that isn't directly created on the first layer of a blockchain. It's not produced through mining. Rather, a token is created by a "smart contract" which is basically executable code running on top of a blockchain. You can basically write some code in Solidity, (Ethereum's programming language) to create a new token, then distribute these tokens by trading/selling them. Not only can I send Ethereum to other ethereum addresses over the network like bitcoin, I can also create, then send these tokens to other addresses, and the transaction is recorded on the blockchain for eternity since Ethereum allows executable code AKA "smart contracts".
Fungibility in this context basically references a token's, or financial unit of value's unique characteristics. If there are no unique characteristics from one token to the next, other than it's chain of custody recorded on the blockchain, it's considered fungible. Fungible is basically another word for interchangeability. Fungible tokens hold the same value one token to the next. Basically, if given a choice between trading for two fungible tokens, their value is identical, and the person trying to obtain one goes for the lowest price every time because there's no difference between the two.
Non-fungible means you have a token with no equals, or identical counterparts. Think of it as a gold medal, inscribed with the event an athlete won like "The 2021 Boston Marathon 1st place". There can be only one first place winner, and 2021 only happens once. So that gold medals value can take on a not only it's weight in gold, which is a fungible aspect, it also takes on another layer of value, it's collectibility. Let's say you have a choice between purchasing a Jesse Owens 1936 olympic gold medal, and a 2020 olympic gold medal from a nobody that will never be famous, would you even care how much actual gold was in each? Fungibility, and non fungibility are two economic distinctions that provide value for two separate reasons. One, it's universal interchangeability which lends itself well to currency, the other, it's unique characteristic as a collectible.
In Practice
So basically, and NFT is just a digital token that is unique. The smart contract that was used to create it can easily serialize it with a specific serial number, then on their own web site have artwork assigned to that serial number. In most cases you'd log in using a metamask wallet, and you're served up the artwork by their server for whatever NFT's the address you're using owns. In some cases, the actual artwork itself like the code to render a jpeg, is part of the contract, forever accessible on Ethereum blockchain itself.
My first experience with NFT's was collecting Crypto Kitties back in 2017. When they launched the site, kryptokitties were so popular it made the ETH blockchain almost unusable, and drove the fees for getting a transaction validated through the roof.
You basically get a serialized "token" sent to your metamask address, and that's how you sign into the site, and validate ownership. You can only create new cryptokitties by "breeding" them with existing cryptokitties so you get a royalty from the sale of newly minted cryptokitties if you've sired a new one.
This brings us to the 2021 style NFT's for artwork that are all the rage now. Basically you take artwork, assign it to a smart contract with single ownership properties to the token and boom, you've just created an NFT. Biggest overlooked attribute by people who don't get the value aspect is that it's uncounterfeittable. This makes it even more collectible in some cases than an actual physical work of art, a rare baseball card, or your favorite rare pokemon.
You can even assign royalties to future sales to go to the artists's address. Like if the NFT you create gets sold by one party, to another, 10% of the sale goes to the original creator's address. A successful NFT artist could create value for generations. Imagine if DiVinci's descendants got a percentage of every sale of his work.
This concept would normally be considered "ahead of it's time", but technology is advancing at an exponential rate, and it only took Ethereum 5 years after being invented for these things to catch on. If it's just a trend isn't even a question. People thought the internet was a fad, they thought bitcoin was a fad, they even though cryptokitties was a fad but mine have only gone up in value. What most people picture is a physical work of art's ownership will be tied to an NFT in the future. We just need courts, and the general public to catch up for that to happen.