Yeah, you're missing a crucial step. Assuming you patched your games with the Supercard patching software, after saving in-game you must press the L+R+A+Select button combo in order to send the updated SRAM to the *.sav file on your card (a menu should appear). This is outlined in the Supercard website's FAQ section.
For games you haven't patched (homebrew for instance) you can use the QPC Method of saving, which stands for Quick Power Cycle: save in-game, turn off the GBA, quickly turn it back on, press Right to get to the saver tab, navigate to the *.sav file, and press A twice. This will manually overwrite the *.sav file with the SRAM that's still saved since you powered on before the data was lost.
"Extracting" save files from the Supercard is as simple as copying and pasting them from your card to the computer.
Note that while the Supercard does contain a built-in version of Goomba Color so you can launch *.gbc files directly from the menu, this version is outdated, so you'd be better off creating your own Goomba Color compilation manually and sending it through the Supercard patcher (it's one of the few GBA homebrew titles that need patching).