The provider puts the store on the React component context. You then use the @connect decorators in your component definition (in this case Layout) to connect to the store and select the data you need.
The connect accesses the store from the context and subscribes to changes, so that every time the data in the store changes the component is re-rendered. You should never pass the store via props, it's bad practice. You can pass the connected properties down to children of course, but never the complete store.
The initial data that you put in the store will be rendered the first time your app mounts to the DOM. Every change thereafter (user form submit) will mutate the store and your component will update through the connect mechanism.
The Redux docs are well written. I suggest you take a closer look at them. There's also plenty of tutorials about Redux that are free, for example on egghead.io.