Everything gives you feedback of what's happening in the System Console, so you might want to open it when using it to see what exactly it's doing.
Applying shape keys
When you click Apply the addon will read all the shape keys from the selected object, save their values and then set them all to 0. Once this is done it'll reiterate from the selected shapekey in order down the list up to the shape key you set as bookmark, for this you need to type the EXACT name in the first input field ("Apply shapekeys from selected to:"), it'll ignore the rest of the shape keys. For every shape key in this list it'll be as if you created a "New shape key from mix" with the saved value from this specific shape key (since every other shape key is with a value of 0) and add as a sufix or prefix whatever you type in the second input field ("Add ").
Make sure to SAVE BEFORE applying, just in case, you WILL lose the keyframes you had in the applied shape keys because it is deleting and creating a new one. There will probably be an update in the future that saves your keyframes to the new shape key.
What do I use this for? The most common use I can think of (and what I created it for) is for creating variations of a character using shapekeys as controllers and then applying them.
Renaming shape keys
This works just as you imagine it would. Type what you're looking for (caps and spaces matter) and it'll automatically replace it with what you type in the second input field.
Deleting shape keys
You can delete shape keys by selection: from your selected shape key to the last shape key matching exactly what you type in the bookmark input field (which appears once you tick on "By selection?"); or just by looking for something in the name (either it includes or does not include whatever it is in the "this in name:" input field).
Both methods work together, so you can look for something with a specific name in just the selection as well. If you leave the "this in name:" field empty then it'll delete everything (or just selected if you make a selection).
Moving shape keys
You can move a selection of shape keys a specific number of slots either up or down it's current place in the list or directly below another shape key if you choose "Bellow..." in "Where to?", the "How many slots?" field will change for you to input the EXACT name of the shape key you want to use as a bookmark to move the selected shape keys.
The Apply Shapekeys part starts at 3:50
If there's any questions please send me a message and I'll answer as soon as I can.