The overwhelming majority of Chinese characters are composed of simpler characters. That helps a lot. You don't even have to be told that, you'll figure that out yourself fairly quickly. Being taught what the typical components are (there are several hundred) doesn't seem to be a shortcut, but you will need to roughly know them in order to use old-fashioned paper dictionaries.

Most characters have a sound part and a semantic part. The sound part is not very precise, but it helps. The semantic part can be quite abstract, such as the sign for mouth (a square or a squarish rectangle) for parts of speech (和 = and).

Like the others wrote, a phonetic system is used in the beginning to provide the pronunciation to the kids. The same system is usually used later for text input on computers or cell phones, possibly supplemented with support for drawing characters.

They have the additional problem that they might not speak Mandarin and the pronunciation support they are using is based on Mandarin.

It works much better than it has any right to, but it requires much more training to reach basic literacy than even an imperfect sound-based system like English. Weeks versus years. To reach proper literacy takes years and mountains of text in both cases.