How are meteorologists able to forecast the local weather 10 to 15 days out?
They can't. Their long range forecasts are 7 days, at best and are mostly based on computer generated probabilities. They do a remarkably job, given all the variables of our atmosphere.
Computers. Vast amounts of data are input to a computer, including current conditions over a wide area, historical data, etc. Then the computer crunches the numbers based on an algorithm and spits out a forcast. They've actually became fairly accurate the last decade.
looking and measuring wind currents, with radar
Satellites, and before that, observations about what preceded an event...