TL;DR

The ebooks of the Amazon Kindle app can be found on your Android phone in PRC format below the folder /data/media/0/Android/data/com.amazon.kindle/files/.

Download the books

To be able to extract your books from your Android phone, you need to synchronise the books first, so they are available on your device.

To ensure they are local. turn of all network connections (wifi, mobile data) and try to read them.

Finding the files

At first I guessed the files I was looking for may be ending in .azw. Looking for these files, however did not reveal any file.

My second guess was to look for amazon files or folder named amazon:

% find / -name \*amazon\* 2>/dev/null

I found /data/data/com.amazon.kindle, with the following content:

% ls
app_com.amazon.device.syncHIGH    app_metricsNORMAL                 databases
app_com.amazon.device.syncNORMAL  app_web_cache                     files
app_dex                           app_web_database                  lib
app_dexopt                        app_webview                       shared_prefs
app_metricsHIGH                   cache

Looking for books in this directory wasn't successful. However, the databases directory looked interesting.

% cd databases
% grep -ri mybookname *
Binary file databases/kindle_library.db matches
Binary file databases/kindle_library.db-journal matches

I used sqlite to have a look at the database:

% sqlite3 kindle_library.db
sqlite> .schema
...

This revealed one interesting table named LocalContent. Looking at it closer:

sqlite> select * from LocalContent;

This revealed the emulated path /storage/emulated/0/Android/data/com.amazon.kindle/files/ and referenced .prc files.

I looked for them in the filesystem using ...

% find / -name \*.prc 2>/dev/null

... and found my books!

To extract the ebooks to your computer, you can use any file copy program. The rsync utility however is suited for it very well, as it can (re-)sychronise the whole folder:

% rsync -av /data/media/0/Android/data/com.amazon.kindle/files/ mycomputer:mybooks/