when you press the checkout button, Moodle will search for all strings from the core and all plugins and will copy them to the database. The translation tool then works with the strings in the database and the customized strings can be then written back to the files in the local language folders.
The initial checkout when there are no strings registered yet in the database may take a longer time. Later checkouts are usually shorter as only the new and modified strings are updated.
At all sites I tested the new tool at, the initial checkout took reasonable amount of time - up to twenty seconds or so. At your machine, the script was not able to finish the complete checkout within the maximum execution time defined in your php.ini. The fact it took so long at your machine may indicate that your database performance is not very good and you should consider some performance tweaking. Sure, there could be a progress bar or something to indicate what is going on. And the script could even try to detect the upcoming maximum execution time deadline.
Anyway, please try to just re-run the checkout several times. If you have access to your database, try to get the count of already checked-out strings using
SELECT COUNT(*) FROM mdl_report_customlang;
You should see the increasing number of records after each run of the checkout and you need to reach almost 16000 records before the translation tool can be launched.