When you start your tatting, you will have a thread end that needs to be hidden. The easiest and cleanest way to hide it is to hide it as you tat instead of waiting until your tatting is done.
After you tat your first double-stitch, snip the thread end to about 1/2" to 1" long. Fold it back behind the double-stitch and parallel to the shuttle thread. Then tat over it another 3-5 stitches and snip it close to your tatting. This provides a smooth, strong connection. I've never had any of my tatting come apart when I use this method of hiding the starting thread ends.