Preventing swing set legs from sinking
I Assume your set is like the one in the video and has wood 4x4 legs and that you have cut the correct angle on the bottom of the leg so it is flat to the ground.

You could cut a square piece from a 2x6 ( or 2x8 if you like, treated or outdoor wood ) and attach it the bottom of each leg.  This would spread the weight out over a broader area. IF you leave them on top of the grass You could use a router or a saw set at an angle to round over the all the corners and edges to eliminate all sharp right angles both top and bottom ( top for safety and bottom so you can drag it to a new location without the edges digging in the grass ) With this method the base/foot will travel with the set if it get moved.
But i would recess them into the ground to avoid a tripping hazard.

Another possibility is buy a some pre-made concrete pavers but these are very hard obviously and i would recess them into the ground and glue foam padding all over them or cover with a layer of sand. You do not want a kid to fall and hit their head on them ( or the wood for that matter.) This option would not allow for moving the swingset with out re-setting the pavers. 

Or go full  child safety and do like they do on school playgrounds and anchor the set deep and put down sand or shredded rubber to lesson the IMPACT when child falls of the swing.