On Noteflight, the shift key plus either up arrow or down arrow will transpose the selected elements up or downwards respectively by a chromatic semitone. This can be used in conjunction with the key signature change to transpose the music to a new key with a new key signature.
The example passage is in the key of C major (or whatever key the no-sharps-or-flats signature happens to be representing here). When changing the key signature by adding 3 flats, you've changed the key signature by three semitones (coincidence) from C to Eb major. Your key signature is written in Eb, but your music is still sounding in C since the change key signature function doesn't move any of the notes in Noteflight. In order to get the correct notes in Eb, simply select the entire passage, then hold down the shift key and press the up arrow three times. You may need to use the z key as well for enharmonic respellings of notes (D#=Eb=Fbb, et cetera), especially for chromatic notes.
There may also be a specific transpose feature built into Noteflight that does what the question asks, which I leave as an exercise to the reader to go research.