After studying for another two months (and still going :)) ), I now think I know my mistake. места is singular, so it would translate as space/open area. Seats (or places) would be мест which is plural. So lots of seats/places would be много мест. A native could correct me if I'm wrong
To puzzle through this:
The preposition Много takes genitive case for it's object, see: https://forum.duolingo.com/comment/28544274
The nominative neuter singular for "place/space" is место. The genitive singular of место is места, see https://forum.duolingo.com/comment/29038061
thus "much/a lot of space" = много места (genitive neuter singular).
If you look at the noun-ending declension table in the link above, you can see that мест is the genitive neuter plural of место.
You can also see that места is the nominative and accusative (inanimate) plural of место. But the reason you know it's the genitive singular is the fact that места is the (genitive) object of the pronoun много.
So, the most important and first step is determine what case the object of a preposition takes.