This context information comes from various language support plugins, which
can be used after a given file has been saved (so it can check the
filetype and use the correct language support).
KDevelop's completion is set to appear as you type, right away, pretty
much everywhere that it could possibly complete something. This is
configurable in Settings -> Configure KDevelop -> Language Support. If it isn't already set (as it should, by default),
make sure Enable Automatic Invocation is set.