Upgrade to a current version of Fritzing. 0.9.3b is 7 years old and has many bugs (especially in gerber processing.) Current version is 0.9.10 and appears to work more correctly (although things may still be wrong for a variety of reasons such as bad parts.)
0.9.3b gerber output (which indeed looks incorrect!)
0.9.10 gerber output (more correct!)
edit:
You indeed look to have part problems. The HM-TRP part looks incorrect and DRC (routing->design rules check) is complaining about a variety of overlaps. In this image the red squares on the pads indicates that DRC thinks they are too close (which likely indicates the part is wrong.)
FritzingCheckPart.py may help you to figure out what is wrong with the imported parts. This tutorial may help as well
For example (possibly one of many!) the hm-trp transciever module appears from the data sheet to be 2mm headers but the size of the holes and pads is wrong. The header on the left is a correct 2mm header, the right is the hm-trp part
the hole size should be 0.032in not 0.049in
; NON-PLATED HOLES START AT T1
; THROUGH (PLATED) HOLES START AT T100
M48
INCH
T100C0.031500
T101C0.049654
%
so you likely have some work to do before this will work correctly.
Peter