Historically ANY was a token that could be earned through using AnySwap. There have been various ways in which it could be earned, such as providing liquidity, swaping specific pairings on AnySwap, submitting various AnySwap LP-tokens into specific farming or staking contracts. It could also be received through helping to test out AWN and DCRM-nodes and through help with promotions/marketing or helping to keep the AnySwap community in order. Currently these ways should be regarded as the "initial distribution" and are not likely to return as ways to earn ANY.
ANY is a cross-chain asset which has its native contract on the Fusion blockchain and was the first ever FRC20 token to be released on the open market (which was done on AnySwap itself at the same time as it came into existence).
The value in ANY is that it holds governance power over the future of AnySwap. If you have 1000 ANY on AnySwap Fusion outside a farming contract, you may create a proposal on https://vote.anyswap.exchange/ which will be taken under advisement of the AnySwap development team (especially if there's a lot of votes). Often though, the easier a proposal is to implement, the higher priority it will get. Development resources and founder visions needs to be balanced with user desires. As such the result of a community initiated vote may not always be final. Voting instructions can be found here. Voting was mostly used for deciding new listings and new bridges, but after the launch of multichain.xyz, listings have been free and unnecessary in terms of passing a vote. Thus governance value, though still in effect, has not seen active use for quite some time.
ANY should currently be regarded mostly as a "company token". A bet on ANY is a bet on the success of AnySwap and that the AnySwap company does what it can to increase the value of its ANY token. It's somewhat in AnySwap's interest to keep the value of ANY high, as it helps attract use of the platform, and brings more profits for AnySwap. AnySwap has various ways in which it can earn other crypto besides ANY. The primary way is collecting quite high fees when handling cross chain transactions through the AnySwap bridge, but also whenever someone swaps on AnySwap a tiny 0,1% fee is collected by the AnySwap company. Overall it has a few set measures to collect fair fees from users and a wise use of those profits is to maintain the value of ANY as to attract even more users and profit. The case for ANY here is, that AnySwap can use part of its profits to boost ANY. Recently it's been announced that AnySwap funds will buyback and burn ANY with a large portion of collected fees, making the tie between the success of AnySwap and the value of ANY much more direct than previously.
Many also consider ANY to hold value simply because AnySwap is merging chains together in ways nobody else has done before, exposing ANY along the way in a top position across all swaps on all chains. And indeed to get the best liquidity pairs on AnySwap in cross chain interaction, the route will often be through ANY itself.ANY is right now also actively pushed as a an asset adopted by other DEX protocols who benefit from AnySwaps cross-chain bridges. ANY therefore finds itself being a way to earn yield on more and more DeFI protocols. Theoretically this phenomena may end up increasing the value of ANY as more and more chains are added to AnySwap, more and more protocols may wish to bind up some ANY.
Buying ANY is almsot exclusively done through AnySwap itself or other DEXs, making it a little exclusive to the DeFi world (though it does also have a couple of CEX listings, these are not really regarded as important or promoted much).
Providing Liquidity (and farming)
Providing liquidity used to be a primary way to earn ANY, and can still be used to earn by collecting 0,3% fees on swaps. Have a look at AnySwapInfo for details. In many cases the best yields will be in the ANY-pools since these have decent volume and liquidity.
On the Fusion blockchain there used to be an option to stake ANY. This no longer exists, neither do any of the farms. These are all inactivated.
Originially it was planned that AnySwap would be decentralized so that the nodes securing AnySwaps configuration of DCRM will be run by nodes elected by ANY holders. After this happens, those running the nodes might be awarded with ANY. In order to be due for election, it may be required to also hold a certain amount of ANY in staking.
If you want to dig deeper in the current state of ANY and AnySwap, it's recommended to visit AnySwapInfo which holds all information on current supply and detailed info on every pool that exists on AnySwap as well as additional interesting details such as TVL (Total value currently in use through AnySwap). It also hosts a discussion forum where the future of the platform and ANYs tokenomics can be debated. Look also at anyswap.net and multichain.xyz which both hold explorers where you can follow the cross-chain activity.
Lastly, I want to mention that AnySwap tokenomics may still be undergoing revisions, and may be much more advanced in the future than what is described here. A necessary piece of information is that only 20 mil ANY is considered to be open for distribution, the other 80 million is considered to be locked and their fate is only allowed to be decided through the governance where both the AnySwap company and ANY holders can agree. There is no indication that decisions regarding them will happen any time soon.