How does this diode compensate bias current change on a BJT due to temperature variation?
Summarised solution:


Excellent question.
You'll be sorry you asked :-)
Take cup of cofee and deep breath (not while sipping coffe) and read on ...
This is a classic temperature compensation system.
It "sort of works" under selected conditions but is much less effective if not well designed than most people realise, and most places it is used in do not understand it, or design it well, or at all.
Diode thermal characteristic at certain currents relative to Ie is suitable to be used to shape base drive voltage such that variations due to temperature are minimised. Most people trim the diode current empirically. Most explanations reduce to "it works". First ref does a fair job of the theory of why. Other refs give extra insight.
As drawn the circuit is suspect - see below.
(An extra R and slightly different connection is more useful.)




It is usual to return the diode to -Vee and to have a resistor in series in the diode and the resistor is essential for correct operation but as shown has some sort of chance of semi-success. How much you will be able to judge if you wade through the references cited below. It will takea lot of coffee. Operation, or intended operation, is far more complex than may be apparent (how hard can one diode be to understand? :-) ).

The simiplistic description of what happens, and obvious because it is really just a statement of desire, is: 


If diode current is a certain ratio of emitter current then the effect of the voltage variations with temperature in the two junctions results in the base drive voltage being reduced by enough to compensate for the loss in emitter base junction voltage which would otherwise increase transistor drive.


OK. Big deal. All that says is that what it does is what we want.
 The excessively enthused will ask, "Why?"
 They will probably be sorry.

Take a deep breath:

This guy indirectly says Bob Pease agreed with him that it only worked properly when Re was small and diode current so very large compared to emitter current that it would be unacceptable in most cases. Who are we to argue with a man who says that Bob agreed with him ? :-). To read how he puts it look here - Temperature compensation of BJT differential amplifiers and start at the top of page 5. Reading pages 1 to 4 is probably good background  preparation. It is far more messy than needed but not too hard if taken one word at a time with cups of coffee between. The "required diode current is humungous" material is at the top of page 6. 

This Bias circuits for RF devcies explains it more simply on bottom of page 2 , top of page 3 but really says "it works".

This truly magnificent gem really helps BUT fails to close the loop by properly referring back what he is doing to the original problem in a clear and useful way. Skim through this and you'll get some idea. Note the vast range of characteristic curves aas diode types chnge and see why it is as much empirical as tightly designed.
Forward bias characteristics of 1N400x diodes. Don't worry that they are 1N400x power diodes till you have skimmed text.

A paper which could be useful if not having $ asked for is abstracted here

This data sheet is for a Sanken power Darlington with inbuilt temperature compensation diodes. It's description  also reduces to "when we do this then it works" but it gives some useful inight. Note how they have 1 diode in one device and 3 diodes in the other for temperature sensing and then combine the lot plus a series adjustable R. Deep magic. Much coffee ;-).

Another Sanken datasheet

Another paper that could be useful if you have academic access - they use an inverted transistor as the compensating diode.This is for an integrator but would add to the joy if it could be accessed.