As with most things in Eagle, it does not import DXF files natively, but users have written some ULP scripts to do it. The way that these scripts tend to work is that they read the DXF and convert it into Eagle-native draw commands to redraw the contents of the DXF. For some reason, all DXF import scripts that I've found so far just skip importing text from the DXF; all you get is the lines and shapes. In my case, importing a drawing border file, that is enough to get started with.
Cadsoft has a least made it easy to find ULPs by providing hosting and indexing for scripts submitted by their users. If you just go to the following link and type 'DXFimport' into the search tool, you get a variety of versions of this script, starting from an original script and going up to version 1.6 with improvements by several different users resubmitted under the same name and new version levels: http://www.cadsoftusa.com/downloads/ulps
Interestingly, the hint about where to find this came from this post on Cadsoft's own Facebook page concerning version 1.5 of the script: https://www.facebook.com/cadsoftcomputer/posts/556869044353063
Also interestingly, version 1.6 of the script can also be downloaded from github here: https://github.com/erikwilson/import-dxf
This forum post mentions a different script which I haven't tried which might be either better or no longer available: https://www.element14.com/community/thread/16729/l/importing-a-dxf-file-into-eagle-610-steps?displayFullThread=true
Here is another thread where that script is discussed, with links to an article with more detail that I haven't yet tried to follow: https://www.element14.com/community/thread/21171/l/import-dxf-file-in-eagle-pro-620?displayFullThread=true
