The working principle of TypoGento is quite simple. Since TYPO3 and Magento both are PHP web applications they can be integrated on the code level by simple loading one by the other.
In fact, there is a TYPO3 frontend plugin that instantiates the core classes of Magento and makes it render the views needed. Magento itself is changed in a way TYPO3 can control what to be done and what to be displayed.
This mechanism has several advantages:
To achieve these advantages one needs to run the application on a powerful web server. Both applications are quite ressource-consuming and in the end both of them are enterprise solutions that will need enterprise hardware.
The Magento API is a perfect way for ERP or CRM systems to access information about products, customers or orders. Unfortunately, it’s not well-suited for integrating it with another system like TYPO3 in the given scenario. The reason for that is that the Magento API does not allow certain actions like adding products to the cart or sending away an order for security reasons.