What are "linear spatial weightings" and "specific temporal windows" in Philiastides & Sajda (2006)?
I can make a guess, until someone who really knows the answer comes along :) I haven't read the paper and the answer I can give is probably not going to be formal enough for a math student. But I can tell you what I think.

The goal of the paper, I'm guessing, is to look at the pattern of activation recorded by EEG when viewing pictures of faces and cars, and to try to say if the two activity patterns differ. One way to so this is to show some faces and cars, look at EEG activity and tell your algorithm which is which. Then, after a training period, let your algorithm classify future input into faces and cars as well as it can. In the end, you want to see if it can classify above chance. If yes, then you can say with certainty that the pattern of neural activity between cars and faces differs. 

You might then wonder how it differs. If this guesswork is correct, then temporal windows refers to moments in time when this pattern classification works best - for example, 150-200 milliseconds after the picture was displayed. This can tell you something about how much visual processing must happen before objects are seen as belonging to a certain category. If not done properly, though, you might just be detecting simple visual differences such as amount of information in a certain frequency band - not very exciting. A very early temporal window of best performance of your classifier algorithm would indicate that this is the case. In any case, you learn something from seeing where the best temporal window is.

Spatial weightings of the sensors refers to the difference in activity recorded in each of the individual EEG sensors. From that, you might infer that the difference between cars and faces is most prominent in, say,  frontal sensors, which gives you a hint about the possible brain areas involved in category classification. The linear I don't know what to make of unfortunately. It refers to the bit of the algorithm that decides how to classify these spatial patterns, perhaps by relying on how neural activity is spread over a distance (i.e. how activity originating at a certain place will be captured by the sensor just above it, and how by sensors further away), but to know what makes it linear, you would really need to see the algorithm. 

Hope this helps!