I initially found that strange too given that there is a distinct Sri Lankan Tamil dialect (Jaffna, Batticaloa versions etc) but it looks like Sinhala came in as an original Indo-Eurpoean language and mingled with surrounding languages to borrow some words etc. Here are some opinionated commentaries on that:

http://www.applet-magic.com/sinhaleselan.htm

http://www.island.lk/2003/11/05/featur02.html

Would love to see an authoritative source confirming these.

The sinhalese people of Sri Lanka are Indo-European, not dravidian. IIRC they came from Modern day Bengal.

But Sinhalese culture,food etc is closer to Dravidian.