Seili is an island. It can be reached by a public ferry from Nauvo (Nagu in Swedish). The ferry runs three times daily; timetable in Finnish/Swedish (Ma/Må means Monday and Pe/Fr is Friday). The 5-km trip takes ca. 30 min and the ticket costs something like 6 euros per person.
To get to Nauvo, you can take a bus from Turku. If you approach from the direction of Helsinki, you can change into that bus at a bus stop called Piispanristi. Please note that when changing buses, you need to cross the road and walk 100-200 meters ~east to reach the other busstop called Piispanristi. For many of the connections, you can also change into the very same bus even in Turku, which may be easier. Bus timetables Turku-Nauvo (in Finnish) and Helsinki-Piispanristi(-Turku); do check the matching of the bus timetable(s) and the ferry timetable carefully.
You can also arrive in Nauvo by car. Please reserve enough time for the trip: there’s an additional ferry (free of charge) on the way, and you may have to wait for it (up to 15 min). At rush hours, you have to queue even longer. Also note that cars are not allowed on Seili, so reserve time for parking as well. There is a large parking space on the left when you arrive in Nauvo.
You can find some extra information provided by the Archipelago Research Institute.