My new theme contains a link to the above fonts.googleapis.com. Anybody know what it is. If I take it out, will my web site blow up at some future time. My browser can't find this web site

Depending on whether or not you copied all the code there...you could just take it out and see what happens. Make sure you have a GOOD backup first!

I would not do that first though . . .

Read through this to understand Google Fonts, then it might make more sense.
http://designshack.net/articles/css/a-beginners-guide-to-using-google-web-fonts/

That's for whatever Google font/s is used in the theme.
If you remove the link and the text on your site suddenly looks wonky, you'll know why.
They're a great way to easily add fonts to your website, plus, they're free.
https://www.google.com/fonts