After a Wednesday report showed consumer prices rose in January by more than projected, traders used eurodollar futures to express their view that central bankers have a clearer path to tighten. The expected number of increases from now until the end of next year is up to four, from 3.6 before the inflation report.
Yield of Dec19 Future - Current 3Months Libor / 25 bps (1 rate hike)
Libor 3M = 1.84 % Price of Dec 19 Future (Ticker EDZ9) = 97.18 = 2.82 %
Number of hikes = (2.82 - 1.84)/0.25 = 4
Please note these are very simplifying assumptions, as the 3 months Libor is just a proxy on the Fed Funds Target rate.
In my experience Chinese whispers between IR traders and bank/institution strategy/researchers and then journalists is rife. Hikes/cuts are predicted by traders based on FedFund futures or meeting period FFOIS rates. The same goes for GBP or EUR where the OIS rates dictate the probability of hikes/cuts.
Note that your quote didn't directly say the expected number of increases was calculated from the eurodollar futures. It said they were used to express a view. Which is sensible because they are a very liquid product with minimal execution cost. But they are a proxy to bank rate as Lilane states.
As an aside if you assume that the LIBOR/OIS basis is stable over a short period of time and the Eurodollar strip moved 10bps, you can assume the OIS strip moves 10bps also which accounts for 10/25= 40% of a 25bp hike.