From your screenshots, it looks like you are using point features. To get the map you want, you will need to use a polygon dataset containing zip code boundaries.
Then select the Counts and Amounts (color) style.
I used this zip code boundary layer: USA ZIP Code Areas
The example map in your screenshot is actually a choropleth map. They are frequently referred to as a heatmaps, which can be confusing. Standard Co has a nice explainer on the differences: Heatmaps vs Choropleths
- Jen