The latest full version of Raspbian needs a minimum SD card size of 8GB (EDIT: more like 16GB now) to run properly.
If you install the FULL version of NOOBS, this includes compressed installation files for Full Raspbian that takes up about 2GB on its own. Leaving only 5-6GB left of free space to install Raspbian into, which as you are finding is not sufficient.
You have several options:
1) Get a bigger SD card (16GB or bigger)
2) Install NOOBS-Lite instead of NOOBS. This is the same version of NOOBS without the additional installation files for Raspbian. So in this case you would have to connect your RPi to the internet via wifi or ethernet to install Raspbian to your SD card from the internet. You have a Zero W, so you should be able to connect it up to a wifi router if you have one. This will also give you a wider choice of operating systems to install. This avoids "wasting" 2GB of your SD card for the Raspbian installation files that many people complain about, forgetting that this does not apply to NOOBS-lite.
3) You could replace the full version of Raspbian in NOOBS with the standard (cut-down) version, but this is more for advanced users.
4) The main benefits of NOOBS are to install multiple operating systems on the same SD card, or to quickly restore an OS if you need to start again from scratch without internet or another PC available. With such a small SD card, these use cases do not really apply well to you, so you may as well ditch NOOBS and flash Raspbian directly to your SD card using Etcher, as Fruit suggested above.
5) If/when you outgrow NOOBS and get a bigger SD card, try using PINN-lite, which will allow you to install from a much wider selection of operating systems