。
She joined in her children's games.
Literal
She [topic-は] children [genitive-の] play [target-に] joined.
加わる ('to join, take part in') is an intransitive verb taking に for the thing joined — different from the transitive 加える ('to add X to Y'). The に here marks what one joins, not where one is. 遊び ('play, game, fun activity') is the noun form of 遊ぶ ('to play'), formed by the masu-stem nominalization that's productive across countless verbs in Japanese: 動く→動き ('movement'), 笑う→笑い ('laughter'), 始まる→始まり ('beginning').