This is the usual setup for a material with transparency.
You’ve got a transparent shader, wich makes your surfaces … transparent and you have your regular diffuse shader, wich is just a placeholder for any kind of material setup, you’d like to have there … last but not least, you mix these shaders using some sort of mask.
Your texture might not have an alpha channel! Just select it in the Image Editor and check, if there’s this checkerboard texture in the background.
In case of your texture, I think the transparency is “encoded” just with the regular RGB channels. So make sure to switch the “Color Space” dropdown from “sRGB” to “non Color” in your Image Texture node and use the “Color” output instead of the “Alpha” output as a factor for the mix shader.
If you’re using Eevee, transparency isn’t going to work out of the box.
In your Material settings, you need to change the “Blend mode” from “Opaque” to “Alpha Blend”. Otherwise, Eevee is going to treat your transparent elements as solid black to save on performance. If you want more accurate shadows, you might also want to have a look at the Shadow Mode.