Thanks for your questions!

1. Because the atmosphere is a gas, it expands when heated and contracts when cooled. This means that if we identify where the atmosphere is 500 millibars of pressure, for example, the location of that pressure will vary in altitude- in some places it will be higher above the surface (thicker), and in some places it will be lower (thinner). These regions generally correspond to regions of warmer and colder air temperatures, respectively. To summarize, by thickness, we are just referring to the altitude above the surface where the atmosphere is a given pressure. 

2. Nowadays, we use reanalysis data. These products combine satellite and radiosonde (balloon) measurements but assimilate these into a model to fill in the data in space and time so there are no gaps. (see also footnote 3 of this post). However, the very first SSW that was observed in 1952 was indeed measured by balloon, from Berlin Germany.