You set up some resources and just wait and see what the children do with them. I'm still quite a beginner at this idea and you can find a lot more inspiration at the following sites:
The Imagination Tree - info on what an "invitation to play" is and some great ideas and answers to questions.
This Pinterest Board has seriously loads of ideas. It isn't about copying the idea, it's about getting inpiration.
Teach pre-school - some great invitation to... prompts, plus some more links!
Learn with play at home - a great idea for inviting number play.
This week I've tried the following invitations:
A big pile of plant pots, a large tub, and a hose nearby (sunny weather - hose essential!).
The children ignored the tub and the hose, but made castles with the plant pots.
A piece of drain pipe across the swing frame, with a couple of toy boats, and the hose.
The children used the hose to wash the boat down the drainpipe - great fun!
A tray with some poster paints and paint brushes, some paper of different colours, and the pine cones, feathers and sticks which they had collected on a walk the previous afternoon.
The children painted the twigs and pine cones, and have all sorts of ideas of what they will use them for.