Planes can fly in pretty strong winds once the plane gets into the air and pilots are trained for the high speeds of wind to fly in it.
However planes cannot safely take off or land in very high speeds of winds and if the winds are too high then the flight will be delayed for the safety of the passengers and pilots.
If the wind speed is above 60 MPH then the flight will usually be delayed or cancelled until the wind speed drops.
If the wind speed is below 60 MPH then the flight may continue and the plane may go ahead and take off into the air.
You may feel some shaking of the plane going into the air at high wind speeds but it's usually nothing to worry about.
I've had airlines cancel the flights when the wind speed was around 55 MPH and others went ahead with the flight at 60 MPH.
The flights went smoothly though and no problems with the flight happened.