How can I directly manage my files on the sdcard of my Android mobile phone?
Android is a Linux based operating system for mobile devices, that's true. But on the top level all applications are java apps using the Android api for acting in Android and cooperate with other apps. (Unfortenuately?) this is not a Linux/Unix system architecture. Hence (normally) we don't have a bash or other shells for moving, copying or erasing files. And therefore we need an Android based java file manager:
The ASTRO File Manager
The astro file manager is a commercial but cost free, very useful and very powerful application: "it helps you copy, move, delete and rename files". And moreover
" the astro file manager makes it easy to backup and restore public applications and also to install and un-install them. This means you can backup and un-install less frequently used applications, then quickly re-install them when needed. Since Android does not allow applications to be run from the SD card, this offers you the next best solution for freeing up the limit internal memory. "
As Android application the astro file manager is recommended by several reviews - e.g. by androidtapp or by androidpid. Hence although this application is 'only' free in the sense of freebeer it's not only a helpful tool but also a very well designed piece of software.
The Linda File Manager
The linda file manager should be listed too, because it's another well known alternative - although it's also "only" freeware, not open source software.
And a Really Free Open Source File Manager? - have a look at OpenIntents!
A very well elaborated and nicely designed file manager is the very useful and really free open source OI file manager. It's generated by the project OpenIntents and published under the Apache 2 licence: With the OI file manager you can do what ever you expect a file manager should do.