What you're seeing are the ARTCC "Points of Communication" - their transmitters/receivers. Use the closest one (1 on the drop down list) to contact them. The bearing is their bearing from you. i.e. Bearing 090 means the station is east of you.
Here's a map some guy with too much time on his hands made of the ZFW sites.
FMI: Pg 19 here:
http://www.garmin.com/manuals/GPSMAP196_PilotsGuide.pdf
If you want flight following not in Center's airspace, you can use the NRST and then choose Airport instead.
Or you can use the cursor to put your pointer inside the nearest airspace. The airspace borders will turn dark gray. Click enter. You'll have a screen and on the top it will say what class of airspace it is. The 2nd line down is "Controlling Agency." The default button is "OK." Click left once to highlight Frequencies. Press up once and that will put you in the frequency list for the controlling airport. If there is more than one (mine has 125.00 and 127.20), they will have a * beside them which means "usage restrictions." Click Enter on the frequency and it will give you the sector in which to use that frequency or other restrictions.
Clear as mud?