In addition to previous answers:
The easiest way of proving that space is Baire is using one of following theorems:
1)Any locally compact space is Baire 2)Any complete metric space is Baire
In fact, there is a notion of Cech completeness which generalises both theorems. (A space is called Cech-complete if remainder of its Stone-Cech compactification $\beta X\setminus X$ is a $F_{\sigma}$ in Stone-Cech compactification, every locally compact is Cech-complete and every complete metric space is Cech-complete).
Then, while product of Baire spaces need not to be Baire, the product of ANY(even uncountable!) collection of Cech-complete spaces is Baire.