Two ways: ferry or small plane.
Ferry — The passenger and car ferry runs from Ceiba (the new terminal on the east coast of Puerto Rico, replacing the old Fajardo dock) to Dewey. Crossing takes about 90 minutes. Tickets are cheap but the schedule is unreliable and queues for the car ferry are long; locals and residents get priority. Reserve online if you can, and arrive early.
Plane — Air Flamenco and Vieques Air Link fly from San Juan (SIG and SJU) and Ceiba (RVR) to Culebra's Benjamín Rivera Noriega Airport in nine-seat Britten-Norman Islanders. The flight takes 25–30 minutes from San Juan. Bags are weighed strictly; expect a $1/lb fee over 25 lb.