Does a low R-squared value in climate temp data mean that it can't be used to prove climate change? 
I have this climate data, it has a low r-squared number - does that mean it fails to demonstrate climate change?

No, this model with this dataset does not contradict climate change.
I believe you're using monthly temperature series, which means the sample size is about n=2,000. Your correlation coefficient is $r=\sqrt{R^2}\approx0.09$, the t-stat is $t=r\sqrt{\frac{n-2}{1-R^2}}\approx 4.2$. Yes, correlation is very low, but it is significant at 1%.
Does this mean that this model and dataset prove that climate is changing or that the change is caused by humans? No. That's much higher burden of proof.
However, from what we know outside this dataset, it would be strange if the temperature wasn't changing over long periods time now when it was clearly changing in the past before humans even existed. Climate is always changing. The question is why and what to do about it if anything at all?
