Sometimes a good way to think about these things in the future can be to take them to the extreme. Consider your climb speed is about 70kts and you are climbing at 1000fpm. If you had a 70kt headwind then you would be climbing vertically so more or less hovering over one spot on the ground. Now after 1 minute you are at 1000ft and havent moved an inch so your climb angle is absolutely vertical!
Lets change that now to a 70kt tailwind. You are still climbing at 1000fpm so after 1 minute you are still at 1000ft therefore your RATE of CLIMB is exactly the same. Your angle however is far flatter as you have been travelling forwards at 140kts for that minute which has taken you well away from where you began.
P.S. The wind moving left to right and vice versa is not relevant in a question like this until later on in your studies when you will learn about effective TAS.