Largely, microphone technique.
They used relatively few of them, positioned the band and singer in the studio to work well with that style of mic-ing. Everybody could hear everybody they needed to - the rhythm section would be closest to the piano and singer, the loud blowers furthest away. The unit was inherently self-balancing; no electric instruments to mess it up - or at least, if there were (say, electric guitar) the amplifier volume would be sent to blend correctly into the mix.
Musicians would stand and play louder for solos, thus bringing them out in the audio picture just enough to work; there are many examples from this era of this technique being over cooked - many Stan Getz records exhibit this type of problem.
The studios had a sound that lent itself to the style of music; not too dead, not too live - just nice. Some studios were considered so special they wouldn't even risk painting the walls for fear of messing up the sound. Fundamentally, everyone was there to capture a musical performance.
The difference today is nothing to do with technology; back then, record producers were like photographers. Today they have become more akin to sculptors.