If you are flying to the festival, the airport you are looking for is Amsterdam Schiphol, the main airport in the Netherlands. The train station Leiden Centraal is only 15-20 minutes away from the airport (trains to Leiden typically depart from platform 5 or 6 at the airport). A bit further away from Leiden, but definitely reachable (about 1h with public transport), is the Rotterdam The Hague airport. Most flights to and from Rotterdam The Hague airport are operated by Transavia.
Leiden is easily reached by train. It is only 10-15 minutes from The Hague and the journey from Amsterdam takes around 30-40 minutes. Ideally located between Amsterdam, The Hague, Utrecht and Rotterdam, Leiden is a great spot to be if you want to match the festival with a visit to the nearby cities.
The main station is Leiden Centraal, on the north side of the city centre. Leiden Lammenschans station is on the south side, about 1km, away from the town hall and easily connects Leiden to Alphen de Rijn and Utrecht.
If you are coming from further away, there are options for you too! For example, the Thalys easily connects Paris, Lille, Brussels, and Antwerp, to Amsterdam; and the tickets are quite cheap if bought in advance. From London instead you can get an Eurostar.
The highways A4 and A44 pass close to Leiden. However, travelling by car in the city centre might be difficult at times, due to the number of buses and bikes circulating relentlessly. Your best bet is to park your car just outside the Singel (the canal surrounding the city centre), and move in the city centre walking, by bus, or by bike. Keep your eyes open for park meters as most parking spots are not free.