Import Gerber files into Altium
I have downloaded the design you are dealing with. The gerber files, all layer files have .PHO extension.





Import gerbers into Altium

If you rename the files as follows you will be able to open them in Altium:


Layer4.pho \$ \longrightarrow \$ Layer4.GBL,  this is the bottom layer
Layer1.pho \$ \longrightarrow \$ Layer1.GTL, this is the top layer
Layer2.pho \$ \longrightarrow \$ Layer2.GP1, GND plane 1
Layer3.pho \$ \longrightarrow \$ Layer3.GP2, GND plane 2
SoldermaskBottom.pho \$ \longrightarrow \$ SoldermaskBottom.GBS
SoldermaskTop.pho \$ \longrightarrow \$ SoldermaskTop.GTS


I have identified the layers by the PDF appendix.
For Pastemask (GBP, GBT) and Silkscreen layers you should search for their Altium extensions. 

These modified files can be dragged and dropped into Altium, example for Top layer:



This way all layer files are easily openable in Altium.



Export TopLayer gerber into PCB document file


Select Layer1.GTL tab in Altium
Select the Menu → Tables → Layers Order option. Following dialog will be shown:
Fill the cells as below.


Enable the Export to PCB option as follows: Menu → Tools
→ Netlist → Extract.


  "After a netlist has been extracted from your CAM data, the File »
  Export » Export to PCB command becomes enabled" source

Now, select Menu → File → Export → Export to PCB and a new PCB document will be opened with the Top Layer. 

Result: