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?

Let's review what $R^2$ means in this case.

Proportion of variance explained by the model.

What your value says is that the model explains $0.86103\%$ of the variance. This does not surprise me. Climate change is talked about as being a few degrees over decades. It is December. I suspect that, six months from now, it will be a lot more than two degrees warmer. Many factors other than "climate change" would affect an average monthly temperature. If I wanted to predict an average temperature for a month, I would find it much more helpful to know that the month is in June than the year is 1880.
(In other words, which do you think is hotter, winters now or summers 140 years ago?)
No, the low $R^2$ value does not preclude a trend.
Your next step in the analysis would be to look at a confidence interval on the slope parameter. If that confidence interval contains numbers less than $0$, then, loosely speaking, it could be that your positive trend is just by coincidence, despite the true trend being cooling.
