On-chain analysis goes by several names. It is sometimes called blockchain analysis, on-chain analytics, blockchain economics, or network forensics. Whichever name is used, a similar analysis is taking place.
Data created by transactions that take place on a blockchain is used to understand the behaviour or motivations of the participants of that blockchain.
Let's look at an example. When Bob sends Alice one bitcoin, that transaction is recorded on the Bitcoin blockchain. We are able to look at that transaction data to see from which wallets the bitcoin was sent from and to, the amount of bitcoin that was sent, and the time it was sent. Because all transactions are recorded on the blockchain and cannot be changed, we also know all the historical transactions of that one bitcoin since it was mined (created). So, not only can we see the data from the recent transaction of Bob sending the bitcoin to Alice, but we can also see data for all the previous transactions of that bitcoin before it came into Bob's possession.
By grouping and analysing this publicly available blockchain data it is possible to learn about the actions and motivations of market participants over time. Price data can then be overlaid on top of this blockchain data to provide useful forecasting tools such as the ones found on this site. For example, by understanding when Bitcoin miners are generating large amounts of profits from their on-chain mining activity, it can give us clues as to whether the market as a whole may be overheated from a price perspective. See the Puell Multiple Live Chart for more on this.
This is a very topline explanation of what blockchain analysis is. This Investopedia article by Jake Frankenfield goes into some more detail about on-chain transactions. I would also recommend reading the articles hyperlinked underneath the relevant charts on this site.
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