Moving the creation of the jpgFilter into the setup method and setting the accept method public solves it. 

The main problem is that global variables are initialized before setup() and draw(), however only after setup() is the sketch aware of it's path. So that's why loading of assets that are dependent on the sketch path should be done from inside setup().

Adapted Code

1. import java.io.*;
2.  
3. String[] filenames;
4. String filterExtension = "jpg";
5.  
6. void setup() {
7.   filenames = getFilenames("images");
8.   if (filenames == null) {
9.     println("Folder does not exist");
10.   } else {
11.     println(filenames.length + " " + filterExtension + " files in specified directory");
12.     for (int i=0; i<filenames.length; i++) {
13.       println(filenames[i]);
14.     }
15.   }
16.   exit();
17. }
18.  
19. void draw() {
20. }
21.  
22. String[] getFilenames(String folder) {
23.   return new File(sketchPath(folder)).list(filter);
24. }
25.  
26. FilenameFilter filter = new FilenameFilter() {
27.   boolean accept(File dir, String name) {
28.     return name.toLowerCase().endsWith(filterExtension);
29.   }
30. };

Thank you both, I found some more applicable code for what I am trying to do in the meantime but this helps my understanding!

Although it's generally a safer bet to instantiate things inside setup() for Processing,
when we're trying to understand what's under the hood, we gotta be more adventurous! 

As amnon.owed stated, the "sketch" isn't aware where its path is until setup().
So we shouldn't try to read that type of information too early.

Same is true for width & height. If a class' constructor reads any of those variables within it,
we can't instantiate that class before setup().

Now, putting apart those 2 specific cases above, I had no problems instantiating classes early! 

Moving the creation of the jpgFilter into the setup method and setting the accept method public solves it.

The jpgFilter stores the anonymous instantiation's reference for a FilenameFilter.
However, that's actually gonna be used much later on. So no problems instantiating it outside setup().

Processing's pre-processor automatically inserts the keyword public in our codes
before turning them into valid Java codes.

The only exception is for anonymous instantiations. The pre-processor isn't smart enough to put public there.
However, if an anonymous instantiation is outside any methods, the pre-processor figures it out! 

And by the way, here's a recent post dealing with FilenameFilter & selectFolder(): 
http://forum.processing.org/topic/listing-last-10-modified-files-in-directory

General advice: take a look at the FAQs of the wiki, like the Technical FAQ (the second article is relevant here), it can help you later...