OS X 10.9.5

Open App Store on your computer in the Applications folder. Put "Install macOS High Sierra" into the search field to locate the installer. Check to be sure you are downloading the full installer - around 5GBs. Download the installer which will open automatically when it has finished downloading. Proceed with the installation. Also, see How to get macOS Sierra - Apple Support.

Before you do the above here are some useful preparations to help with a trouble-free upgrade: Basic Steps for Upgrading OS X. Run First Aid then make a fully bootable backup.

You can run High Sierra. Download it from the App Store.

macOS High Sierra on the Mac App Store

Check your Applications folder for "Install Mac OS X High Sierra". If the file is there, and complete (~5gB), you can install from there. If not complete, delete it and redownload from the App store.

As always, have a complete, current backup.

FWIW, some people get the 19.5 MB stub instead of the full installer and no one seems to know why. One theory is whatever version of macOS they are running isn't fully updated so make sure you have the latest updates.

Read this article and follow it carefully. It will get you the full installer.

http://osxdaily.com/2017/09/27/download-complete-macos-high-sierra-installer/

You're welcome. It's a 5GB download so if your connection is somewhat slow that would explain it.

Yes, you can delete that particular file. Strangely, I've seen that same 19.5mB figure here on the forum previously.

Yes, indeed.

You're welcome. Glad to help.

Did the installer run? It normally takes 25 mins to an hour. If not, look for the installer in your Applications folder and double-click its icon.

The installer is called "Install macOS High Sierra".