Keep a journal. When something happens, positive or negative, your first reaction may be to call your significant other. Rather than calling, texting or emailing her multiple times a day, write your feelings and reactions in a journal and save your stories to tell her during dinner or on your next long walk. Giving her space may mean just easing up a bit on your contact with her; by not calling her every time something happens, you are giving her a little space. Telling her your stories when you are together keeps her included in your life so she doesn't feel hurt or like she's not included.
Make plans with your friends or family members. This gives her time to herself to enjoy her space while also giving you time to reconnect with the other people in your life. Spending time with others makes you seem less available to her and helps you both appreciate your time together even more.
Talk about how much space she wants. The best way to avoid hurting her is to know exactly what she wants in terms of space. If she just wants you to stop calling her as often as you do, and you mistake her request for space as, "I don't want to see or talk to you for a few weeks", serious problems will occur in the relationship. Ask her if she wants a no-contact order and for how long. If she's unsure what she wants, tell her that in an effort to respect her feelings and space you will wait for her to contact you when she's ready.
Refrain from assuming that because she wants space it means your relationship is over and you are free to do whatever you please. You could do serious damage to the future of your relationship by seeing other girls. Keep the mindset that you are in a relationship and act with the same respect and courtesy you always do.