What's wrong?

Switching language directs to the Home Page

If you want to stay on the same page when switching the language (the same content in the new language) you have to have menu item associations activated. Check the option setting for "Menu associations" in the Language Filter Plugin. It must be set to 'Yes'. If it was 'No' and you have already created sets of translations you can associate all the menu items in Joomla in one go using Synch Assoc functionality. The button is located in the toolbar of the Translations Management Dashboard (version 1.4.2. PRO and higher).

If you want to switch languages at the article (or contact or news feeds) level you have to use the KMFT Language Switcher module if your Joomla! version is lower than 3.1.

In all cases for the language switching to be successful at the item level, the items must be associated.

If you want more information about the Synchronizing Associations process you can watch the video below:

Managing Joomla! Multilingual websites with KMFasTrans