If the client is the one paying, how do the specifics work? For example, how would they trust me to handle their payment details?
You don’t actually take their payment details or sign up to the platform yourself. The client enters their payment information directly into the platform, so you never have access to their credit card. In practice there’s very little risk: the client can always get a refund if you upgrade their plan without notifying them, and you can check the logs to see who accessed the account. As a beginner you don’t need to charge clients large monthly fees; the subscription cost is usually only about 1% of the total problem you’re solving for them, so it’s not a big deal.