The stadium is where you play your official matches. The larger your stadium, the more people can watch your matches and the more money they'll pay for their tickets. Once you have a stadium you will start to build a fanbase. The size of your fanbase decides how many people watch your matches.
Your stadium is an important source of income for your club. If you don't have enough fans however, it will only cost you money. You won't be able to sell enough tickets. That's why you can't upgrade your stadium unless it's been filled for 80% or more in 4 of the past 5 matches.
You can see how many people came to see your matches and the tickets revenue when you go to your stadium (under facilities) and then click the "matches" tab. When a match is at least 80% sold out the attendance number is shown in bold.
If your stadium is among the stadiums with the highest level in your country, matches of the national team can be played in your stadium. As a host, you will receive 33% of the ticket sale for that match. These matches are usually sold out. The cup final and the finals of the continental competitions (champions league, european cup etc) are also played in neutral stadiums.
You can increase the income from your stadium by upgrading it. Not only does this increase the maximum capacity of the stadium but it also increases the price one visitor is willing to pay for a ticket.
You can only think about upgrading your stadium though if the current stadium is reasonable full for a number of matches. You need to have an attendance of at least 80% of the maximum capacity in 4 of your last matches at home before you can upgrade the stadium. When you go to your stadium and click the "matches" tab, the attendances of over 80% are shown in bold.
Only league matches count for upgrading the stadium. The occasional cup match or matches in international competitions can not be used.
Fanshop (needs stadium level 2 or higher)
Catering (needs stadium level 3 or higher)
In addition, fanshop and catering can not get a level that's higher than your stadium. You also can't upgrade any building to level 4 before you have stadium level 1