The openssl part is just using openssl to base64-encode "$@":

  $ echo 'test' | openssl base64
  dGVzdAo=

Which is the same as:

  $ echo 'test' | base64
  dGVzdAo=

The URL is the same as:

  args="$@"
  b64=$(echo -n "$@" | openssl base64)
  url="http://www.commandlinefu.com/commands/matching/$args/$b64/plaintext"

For these purposes though, "$@" could probably be replaced with "$*". Technically the URL is invalid since "$@" could introduce non-escaped spaces into the URL.