The ClientStorageService is an advanced, and highly recommended, option that includes support for HTML Local Storage along with cookies as a fallback or alternative.
Controlling this feature is handled by the idp.storage.htmlLocalStorage property.
Much of that look is obviously controlled by style sheets and message properties, but the "visible" portions are in views/client-storage (to avoid losing your changes on upgrades).
As to why you would use Local Storage, there are really two main reasons:
The main reason for this feature is to enable the IdP's session manager to track and index the sessions created with SPs, and that information does not fit reliably in a cookie. That makes the single-logout feature unusable with client-side sessions unless Local Storage is enabled, since the IdP doesn't know what SPs to communicate with. Enabling the Local Storage feature is necessary but not sufficient to allow at least some form of single logout to work without moving session storage to the server. You also will need to ensure a couple of additional session management properties (idp.session.trackSPSessions and idp.session.secondaryServiceIndex) are enabled, and they are also on by default in new installs. There are two properties because the latter is more a SAML-specific need that may not extend to other protocols in the future.
The consent feature is very limited when cookies are used because the number of records it can store is extremely small. If Local Storage is available, that limit is essentially ignored. If you're comfortable with tracking consent per-device, this is a much more practical way to deploy it at most sites than with a database. Of course, many deployers are not comfortable with per-device consent, but those same deployers may become a lot more comfortable with it after enough database connection failures due to the nearly universally poor quality of JDBC networking code.