A bit late in the day maybe, if so please forgive a newcomer.
An example, I find, always helps to clarify things.
I'm sure many here remember the Carpenters, if not for their music, maybe for Karen's incredible contralto voice.
For those who don't know the terms, an alto sings lower than a soprano (most women's and young boys voices), a contralto (contra-alto) even lower still.
If you take a look at their songbook you'll find that several songs start with an E in the BASS clef! E.g. 'Yesterday Once More' and 'We've Only Just Begun'.
Most women, who'd like to sing either of these songs, would find it impossible to go anywhere near this low.
The result is that most women sing it an octave higher and then find they can't reach the high notes.
Karen's E in the treble clef would result in an E way above the stave.
So what is necessary is to move all the notes of the song up into a range, somewhere in between, where they can sing it.
If, when learning to play the piano, we were good boys or girls and practised all of our scales and arpeggios until we were fluent, then this wouldn't be too much of a problem.
But ...
We've only just begun is in A major and we may need to transpose it up three or four semitones, to make it singable, maybe more.
Er ... ?
So instead, we press the transpose button '+' three or four times, maybe more, and all of a sudden we are playing in a different key, although still playing the same notes that are on the score sheet.
So Karen's low E now becomes a more manageable let's say A below middle C.
So we press E on the keyboard and the instrument sounds the A above, when we have transposed +5.
Bingo! Easy.
The pianist just has to learn how to do that the hard way. It becomes easier the more one practises.
A professional accompanist can do this without any trouble.
Back in the early sixties, my music teacher could do it without, it seemed, even having to think about it as he took the choir practice.
He could transpose any piece into any key on the spot. Mark you, he had been a student at the Royal College of Music.