If you have a Zoysia lawn this is what you want to see! These vines are called stolons. Please do not cut or pull them out! This is one of the ways this type of grass spreads and fills in.
The stolon, often called a “runner”, is growing from the main plant. About every inch on the runner there is a small cluster. These clusters will eventually take a roothold and start to produce another plant. When the stolons finally come into contact with other stolons they will fill in the lawn making the grass thicker and 
fuller.
If the runners start to invade an area you do not want grass, such as, trying to grow over your sidewalk, driveway or in your flower bed, then you would need to trim or cut them. The cut stolon can even be replant in other areas you may want to add more grass too. Be sure the soil is loosened when replanting the runners and the little clusters are making contact with the soil. The runner will eventually take a roothold and start forming new plants.