Helpers is a library, or toolbox, with lots of very useful functions that help you build your modules and sites with joy. It is actually six modules in a single package; you may enable only the parts you need.
A helper would, for example be: number_to_currency(), or distance_of_time_in_words(). Or of course something like select_year(), or even select_country().
Basically all kinds of interesting and extremely useful functions that are too specific to make it into core, but still very handy to have at hand. Stuff that you have wondered that "would be very useful in addition to the default PHP".
How cool would it be to call a $form['country'] = select_country() and get a "select your country" select, pre-filled with all the known countries in the world? Or to have a ready made function to render any number as a currency? Or even to render any percentage in a star-rating piece of HTML (render_rate_as_stars()) etc.
This is part of my plan to "bring more Ruby on Rails delight to Drupal". And most of all to make Drupal a developer friendly environment (where developing no longer requires you to think about the folks that might put the currency symbol behind the value... But where that is simply a case of calling a proper function).