How exactly does MSCI earn money?

MSCI is an "index provider". MSCI makes money from licensing its indexes. ETFs pay MSCI licensing fees based on the assets under management (AUM) and trading volumes of the fund.

It seems to me that I can just look up the list of stocks on the Internet.

Yes, but is the information enough for you to build an ETF that replicates the index? Is it enough for you to understand the index? What if you need detailed historical data? What if you need undelayed price data? What if you need to be immediately notified of index component changes? Suppose you have a mandate to replicate the index as faithfully as possible. Would you feel comfortable with "looking up the list of stocks on the internet"?

According to MSCI's 2020 Form 10-K filing:

Our principal business model is generally to license annual, recurring subscriptions for the majority of our Index, Analytics and ESG products and services for a fee due in advance of the service period. We also license annual recurring subscriptions for the majority of our Real Estate products for a fee which is primarily paid in arrears after the product is delivered, with the exception of the Market Information product for which the fees are generally paid in advance. A portion of our fees comes from clients who use our indexes as the basis for index-linked investment products. Such fees are primarily based on a client's assets under management ("AUM") and trading volumes. [...]

Our business model is susceptible to market movements that affect our AUM revenue, which generally have more influence on our revenues than seasonality.

[...]

Clients use our indexes in many areas of the investment process, including index-linked product creation (e.g., ETFs and futures and options), performance benchmarking, portfolio construction and rebalancing, broker-dealer structured products and asset allocation. We currently calculate more than 226,000 end-of-day indexes daily and more than 12,000 indexes in real time for a variety of markets and industries. Clients receive index data directly from us or from third-party vendors worldwide.

From my understanding of the iShares MSCI World ETF (NYSE ARCA: URTH), the flow of money from this BlackRock ETF to MSCI is something like this (note: I may be wrong):

1. BlackRock has a long-term agreement (if not mistaken, a 10 year agreement) with MSCI for licensing some market indexes. This agreement (apparently named "Index License Agreement for Funds") is probably viewable on SEC EDGAR along with the amendments over the years. (source: news articles, MSCI's Form 10-K filings)
2. BlackRock then sublicenses (?) the MSCI indexes to one of its subsidiaries named BlackRock Fund Advisors (BFA). (source: ETF annual report)
3. BFA is the entity that manages BlackRock ETFs. The largest regular expense of the iShares MSCI World ETF is for "investment advisory fees", which is paid to BFA. (source: ETF annual report)

Important:

If I understand correctly, all of these ETFs own stocks in the same companies, in the same ratio - following a recipe published by the MSCI company.

No, not necessarily. You have to read the ETF prospectus to be sure. Examples of how ETFs tracking the same index may differ from each other:

• Consider the fact that indexes, as published by the index company, do not usually have a "cash" component, while ETFs usually hold a bit of cash. What does the ETF do with its cash?
• Does the ETF need to hold all stocks in the correct ratio at all times, or is there some leeway for the manager to deviate from the exact index?

Read the prospectus; it is part of the due diligence process.