The important thing here is that no vowel appears between “That” and “did”, and between “did” and “the”. This is surprisingly easy, because the sounds are very close in the mouth. When you make the /t/ at the end of “that”, be careful not to let it out. Your tongue is already in position for the /d/, so simply introduce the voice, and release the /d/. This will take practice. Between /d/ and /ð/ again, you need to keep the tongue in contact, but this time you need to change the way you make the /d/ slightly, placing your tongue a little forward, so it comes in contact with your top teeth, ready for the /ð/ sound. This will feel strange at first, and also needs practice. Have fun!
The phrase means “that solved the problem”.