Are apps on AppExchange considered just add-ons to the Salesforce organisations?
Yes. Salesforce is a Platform as a Service (PaaS), so apps are indeed add-ons.
If I have a CRM licence, but install the MailChimp app, the MailChimp app is 'attached' to my organisation, so I can use it, right?
It's "installed," in a way analogous to how one installs software on their computer. If you've used a recent version of Windows, you're probably familiar with the Windows Store, and the four OSes for phones (Google, Apple, Microsoft, and Firefox) also use a similar design model. Granted, it's a pretty rough analogy, but is suitable for our purposes.
It would be comparable for having a software, and purchasing an add-on for it?
Yes. You could visualize it as an operating system and software from the "store" for that operating system, or you could visualize it as a video game that has DLC (Downloadable Content) available for it. The concept is the same.
If a company provides an app on AppExchange, can users just buy the app licence and use it, or do they need to have a platform licence to access the app?
It depends. Salesforce.com has two types of agreement models available on the AppExchange, commonly referred to as ISV and OEM.
With an ISV agreement, the client needs to first purchase CRM licenses, then they can install additional apps from the AppExchange for whatever license fee applies. Both salesforce.com and the ISV receive payments in accordance with the respective license agreements. The ISV pays a percent of their collected license fees to salesforce.com as part of the program. There are two separate bills the client must pay.
With an OEM agreement, the client gets their licenses direct from the third-party, who is provided licenses from salesforce.com directly. The OEM can choose to charge whatever they want per license, and salesforce.com takes a percent of the collected license fees as part of the program. However, the licenses don't include CRM functionality unless explicitly negotiated by the OEM. The client has just one bill to pay, and they are using Salesforce as the PaaS, but without CRM functionality.
For example, if my app uses features available only on Salesforce Enterprise Edition (for example: Integration via web service API), the users are required to have this edition, and purchase the add-on separately. If a customer pays $125 a month to access Enterprise Edition, and the app costs $30 a month, then the pricing goes to $155 per user, correct? It is absolutely not possible for the user to pay only the app fee ($30) and continue using the platform, right?
There's a misunderstanding here. ISV apps from the AppExchange do not necessarily require Enterprise Edition, even if they use the API. Many apps that use the API require only Professional Edition, which would take the Salesforce license down to about $75/month, with an add-on license of $30/month, would make the total only $105/month. The reason why this works is because ISVs are given a special code that, when included in the header of an API call, allows the ISV to complete the call, even if the client doesn't have API access.
Referring to your previous question, however, the nature of the agreement depends on if the app needs CRM functionality. If the app cannot exist without CRM licenses, then the subscriber needs both CRM and ISV licenses (e.g. would be paying $155 from the example question). If the app is an OEM style app, then you only need the OEM licenses unless you also want CRM functionality.
Again, virtually every app out there is an ISV model license, requiring both Salesforce and ISV licenses to run, but there are apps where this isn't true. Multiple OEM packages can be installed in a single org, potentially creating an platform that has multiple useful features, but without leads, campaigns, opportunities, solutions, or cases. OEM licenses do get Accounts and Contacts.
For example, conceptually, you could buy OEM ERP licenses, and gain access to inventory, order tracking, return merchandise agreements, and so on, and also buy a payment processing OEM solution, and you'd have a total warehouse application app that doesn't need CRM licenses to operate. Of course, you lose out on things like forecasting and lead management, but if they're not necessary for an organization's purposes, they could save a lot of money by not buying CRM licenses.