Here is the learning that can take place while a child does crosswords.
Practice reading: the clues are out of context of a story.
Sequencing: doing in order.
Organisation: Doing the across clues first and then the down.
Memory: Remembering which clue s/he is up to.
Reinforcement of the size of letters: fitting them into the square. Having a space for each letter.
Spelling: knowledge about how many letters are in a word. Seeing that sometimes some letters are already in place, so s/he has to allow for that. Develops flexible knowledge about letters in words.
Expanding vocabulary: the example crossword is about rhyming words. Some are about a topic so may be introducing new words. They can be about words that sound the same but have different meanings, or about a words that has more than one meaning.