What is a manifest.json file, and where can I find it on my machine?

There are many json files for your chrome experience from Native chrome json files to json files for chrome extensions. Most of them can be found at

C:\Users\username\AppData\Local\Google\Chrome\

Most manifest.json files are found for extensions in

C:\Users\username\AppData\Local\Google\Chrome\User Data\Default\Extensions

In linux mine are located in .config directory:

./.config/google-chrome/Default/Extensions/bjnkloegafmkhgpjglcbldhaokjpandj/1.0.0.0_0/manifest.json
./.config/google-chrome/Default/Extensions/gbammbheopgpmaagmckhpjbfgdfkpadb/1.9.1.8_0/manifest.json
./.config/google-chrome/Default/Extensions/aknpkdffaafgjchaibgeefbgmgeghloj/1.1.2.1_0/_locales/ar/messages.json
./.config/google-chrome/Default/Extensions/aknpkdffaafgjchaibgeefbgmgeghloj/1.1.2.1_0/_locales/zh-TW/messages.json
./.config/google-chrome/Default/Extensions/aknpkdffaafgjchaibgeefbgmgeghloj/1.1.2.1_0/_locales/es-419/messages.json
./.config/transmission/settings.json
./.config/transmission/stats.json

For the mac it's located here:

~/Library/Application Support/Google/Chrome/Default/Extensions

For further reading: Where does Chrome store userscripts.user.js files on a mac?