If you are using phpBB 3.0.6 or later then reCAPTCHA will be installed by default, so you won't need to download anything.
If you go to www.google.com/recaptcha
and sign-up to reCAPTCHA you should be given a public and private key. Then go to your phpBB forum, go to the Admin CP, go to CAPTCHA modules, select reCAPTCHA and click Configure. You will then be able to enter your public and private keys. Once this is done, reCAPTCHA will no longer be grey in the CAPTCHA modules page, and you will be able to select it as your CAPTCHA module.
Unless your host has disabled networks connections through PHP then it should work fine.