Hi a warm welcome to the site.
Age and stage appropriate means the games are made for a specific group of children so that the activities are understandable, enjoyable, not too easy, not too difficult or challenging and support their development so they make progress towards being able to play games independently and cooperatively. If you check the back of the board game boxes in your setting it'll have information about recommended players ages, maximum numbers and maybe what they rules ar eso you can see what the children benefit from should they play!
Page 138 of this level 2 Cache CYPW handbook explains the ages and stages
4-5 years old/ are able to share and take turns but often need help
5-6/ can cooperate in group play but are not yet ready for team games
6-8/ can play games with rules
8-12/ enjoy playing and inventing games with rules
To research the role and importance of games with rules
page 170 of this level 2 handbook will help - use the 'look inside' feature then search inside the book for 'games with rules'
Games with rules help children to work out social situations, negotiating with their friends to work out what needs to be done.
Rules offer an organised structured form of play, helping children's understanding of order and sequence.
*Games with rules: The Nature of Children's Play
Depending which age group you work with think about the different types of games the setting has. They'll be activities that involve dice, timers, spinners, turn taking, sorting and matching where children are thinking about what they need to do eg. separating the red crayons from the yellow and blue. Games with dice or timers.
Table top games you could review maybe a
puzzle,
card games,
board games eg. lotto,
playdough games (promotional plug there sorry!) matching & sorting, memory games
Hope this helps a little