When you see a star next to a match or a team it means that you can add that match or team to your favourites. The link to your favourites can be found in the top menu.
Adding matches or teams as favourites is similar to bookmarking the match or team page in your browser but does have a few extra features.
When a match is stored as a favourite it means that it will appear in your list of favourite matches. However, a few days after the match has finished it will be cleared from your list of favourite matches so that your list of favourites only shows recent, current or imminent matches.
When a team is stored as a favourite it means that all of that team's matches will be added to your list of favourite matches a day or two before each match starts. This means that you can keep track of your favourite teams' matches without storing each of them as a favourite.