Digital audio recording is easiest to understand correctly if you think of the samples as fractions rather than integers. They're binary fractions, but that's a technicality. The number .25 has the same magnitude as the number .250, but when you see the second as opposed to the first, you have _less uncertainty_ about what's going on in the "thousandths' place". That corresponds to a lower "noise floor" in the purely digital part of each recorder channel--though not its analog circuitry or A/D converter (where any audible noise in a recording usually comes from, if not from the venue or the microphones).
24-bit recording offers you no greater capacity to record higher numeric values (representing greater sound pressure levels) than 16-bit; ±1 is still the absolute outer limit that you don't want to crash into. But up to a point, if your signal path is generally sensible, 24-bit recording can let you reduce your record levels enough so that you don't run out of room at the top, and still not pay a price in increased audible noise. Not always--your mikes themselves and/or your outboard preamp, if you use one, can be overloaded before the signals get to the recorder; again, the ceiling isn't being raised but the floor is being lowered, one may reasonably hope.
The amount that lowering is never anything like what the additional 8 bits would imply (48 dB!), though, because no A/D converter or analog input circuitry has that much dynamic range. One well-known recorder that I measured some years ago gave only <2 dB improvement when going from 16 to 24 under the particular conditions I was measuring it. It wasn't a typical scenario, but it does illustrate the real dependency that exists on all the other factors in your actual scenario. If someone had used that recorder with those signal levels and gain settings except that they thought, "With 24 bits I can reduce my levels by (say) 12 dB to be absolutely safe from overload", their 24-bit recording would then have been ~10 dB noisier than the 16-bit one. Again that is based on an atypical scenario in terms of signal levels and gain settings, but not impossible by any means.
Thus you should always use the input to the recorder and the menu and level settings, if any, that are most appropriate for the signal levels you're dealing with, and always try to get your absolute peak sample values somewhere in the top, say, 6 dB or so of the recorder's range without hitting 0. It's just that with 24 bits as opposed to 16 (and with everything else optimal, please note) there should be no audible "noise penalty" if your highest peaks are only at, say, -8 dB rather than -2. So by setting somewhat conservative levels when you don't know what's going to happen (i.e. life), you're better prepared in the event of the unpredictable. Murphy's Law says, after all, that the unpredictable is the most predictable thing there is.