What is the purpose of the "REF" pin of LTC6915?

where input is (V(in+) - V(in-))

Vout = gain*input + Vref

or to put it another way

(Vout-Vref) = gain * input

or to express it yet another way

differential output (Vout-Vref) = gain * differential input (V(in+) - V(in-))

The reference pin sets the output reference level. This is an instrumentation amplifier with built-in PGA. So ideally the output would be Av((Vin+) - (Vin-))+Vref, where Av is the digitally programmed gain.

Unlike most instrumentation amplifiers, this one uses a flying capacitor topology internally so it has fewer issues with internal nodes saturating (in return you get a bit of subtle weirdness in terms of sensitivity to input resistances and matching of same- see the datasheet), however do heed the equations and limits under "Input Voltage Range".

Like most instrumentation amplifiers, the Vref pin is not high impedance (something like 32kΩ to an internal voltage). For this reason you may wish to drive the Vref pin to something other than 0V.

V– ≤ (Vin+ – Vin–) + Vref ≤ V+ – 1.3

and

V– < Vin+ < V+ and V– < Vin– < V+

There are some additional limits that may apply when using +/- supplies so be sure to read that section.

If you are using +/-5V supplies, for example, you would normally ground the Vref pin. However, for single supply operation you might want to connect Vref to some other voltage within the power supply range. Since the pin (like the Vref pin on most instrumentation amplifiers) is not high impedance you may need a buffer. This is covered in the datasheet.

In the 'Typical Application' shown on the datasheet page 1, for example, the Vref is grounded and a single supply is shown. This has the disadvantage that (since the output cannot be driven below V- = GND) the differential input has to be strictly positive (within a few uV offset) to get any output change. Since the bridge resistors will not be perfect you might also get a dead zone near nominal balance (say zero pressure for a bridge pressure sensor).