The simplest method is to not do anything and let a patch regrow with wild plants. You can remove any unwanted or dangerous plants through weeding, but there is not much maintenance involved with these patches. A wildflower patch that needs to be tended by humans is an oxymoron.
If you want a bit more control over which wildflowers grow in your patch. You can harvest seeds from existing patches of wildflowers and scatter them into your plot. Wildflowers from the dandelion family (Asteraceaa) have dandelion like seeds that you can blow into your patch. Others have small seeds, fruit or pods that you can harvest when the turn mature and brown. Just break the pods and release the seeds onto the surface of your patch.
Transplanting wildflowers is a bit risky since they wilt very fast. Try to not expose the roots of the plants and collect it with its surrounding soil. Make sure that your patch is moist and watered regularly during the first few days to ensure that your plants don’t dry out. Some wildflowers grow by runners and can be planted similar to transplanting.
Not all your flowers might survive, but that’s perfectly fine. All plants require the correct amount of shade and the correct soil type. In other cases some of your plants will be outcompeted by other wildflowers or eaten by herbivores. These are all good learning opportunities to understand the ecology of these flowers better.
Try to allow a mix of different wildflower species to grow in your patch. This will make it more resilient and beneficial to the soil, as well as more useful to wildlife that forage for food in the patch.