Your description matches that of a parasitic vine called dodder. The vines wrap around the host plant and embed themselves into the host’s branches, and then absorb food from the host plant. To control the dodder, remove as much of the vines as you can. You may be able to save the pepper plants from rapid reinfestation by pruning off any parts of the plants that have pieces of the dodder vines embedded in them. If there are pieces of the vine left on the host plant the dodder will simply regrow. In some cases the host plant may be so badly infested that the only option is to pull out the host.

Additional information on dodder from Charlotte County Extension can be found here:

http://charlotte.ifas.ufl.edu/publications/REM-05192009-019--DodderVine-VampirePlant.pdf