Right-click on the button to save it as an image.
Upload the button to your Neocities dashboard.
Using HTML, make the button a clickable link. All you need to do is place the image where you would normally put text.
ezgif.com - an easy-to-use free website for making gifs
flamingtext.com - free word art generator (what I used to make the text effect on the "click here" example button)
Inkscape - free to download vector graphics editor (what I use to design buttons)