Meaning of single trim for BAND-GAP circuits
What does it mean to trim the reference?

When the circuit is made there are all kinds of tolerances on all components. That means that the reference voltage will not be exactly what it is designed to be. You could use only &quot;exact value&quot; components but that could be costly (accurately measure each component) and or impossible (on a chip you do never have exact control of all parameters). So we can implement trimming meaning we make some part(s) of the circuit adjustable. For example a resistor is made from a chain of resistors some of which can be shorted by an electronic switch. Then we can program the circuit to have the correct output value.
Trimming can also be done without programming, for example with a laser-trimmed resistor which is changed in value by a laser such that the circuit is most accurate.

What is the meaning of The spread of Vbe can be corrected by a single room temperature trim?
Why is it precise to trim only at the room temperature?

This bandgap is designed to have a constant output voltage over temperature. That means, at any temperature (for which it is designed to work) the output voltage will be the same. So trimming it at -20 degrees C, + 120 degrees C or room temperature should make no difference. So we will trim at room temperature as that is the most convenient.
Bonus info: an article by the &quot;Czar of Bandgaps&quot;, the late Bob Pease.