You change the key to the interval you want to transpose. With this new key, you can just change the notes a number of spaces up or down and they will all be the same interval.
What do I mean by this? Let me further elaborate trough the use of an example. Let's say you want to transpose a passage that is in C major a Major third up. You use the key that is that interval ( A major third up from C --> E) and then just raises the notes three spaces on the staff.
This will conveniently make all the new notes the desired interval from the given passage.
So, for instance, you have this.
To transpose A major third in the new key you would now have this.
And now as you see all the new notes are a Major third up from the example. ( G -B, B - D#, D- F#, F - A, E- G#, C - E, A - C#, E - G#)
Finally just remember if there are any notes that are raised or lowered they still have to be reflected in your new passage for the intervals to be correct. In other words, don't discard sharps or flats.