What does “care to” mean?

[October wind]

We’re influenced by the happiness of other people in more ways than we care to think of. (Entrance exam of Keio University, the source is unknown. )

Does “care to” mean “want to”?
Why do you think would the writer add “care to” instead of just writing “than we think of.”?

 

[Tarheel]

You could substitute "than we would like to" for "than we care to".

I have no idea why the writer chose those words. (I would say we are influenced by the happiness of others in several ways )

 

[BobK]

...or even 'than we could be bothered to` - which gives a rather bleaker view of human nature