From what I see in your screenshot, it looks like you stumbled upon a Craft 3 installation, in which case neither a Craft nor Public folder exist.
Craft 3 specific
The public folder is now called web, so that's where you should point your MAMP documentroot (or whatever that's called in MAMP). The Craft code is now located in composer's vendor folder, that should magically appear after running composer install. Craft's source would then be in vendor/craftcms/cms.
You can read more about the overall structure of a Craft CMS 3 project in the Setting up a New Craft CMS 3 Project article.
Frontend build tools and JS dependencies (package.json, webpack.mix.js)
Further more, I see that Laravel Mix is used for frontend (js, css) automization, meaning that you have to install JS dependencies by running npm install or yarn install, after which you can use the command defined in package.json to run Mix. I guess this is something like npm run watch (when developing) or npm run prod before you push anything to production.
Modules
And: I see a modules folder, that would most likely mean that a custom module was used for this specific site. Modules are a (poorly documented ;)) Craft 3 way of extending your site without building a plugin and trying to get that to work through composer. They are essentially a wrapper for Yii2's modules.
It's very possible that the Module is just vestigial, however, since that folder is created by default if you do composer create-project -s RC craftcms/craft PATH. You can read more about Modules in the Enhancing a Craft CMS 3 Website with a Custom Module article.
Environment settings
Last but not least: copy .env.example to .env and use that to set credentials for your environment (db settings and stuff).
