The notation 'Ast = N16@175mm centres' means 16mm diameter bars spaced at 175mm centres (as per his cross sectional diagram). 'N' would be the type of bar which would depend on the country/code/standard implied in the question.
To calculate the area of steel in a 1m wide strip of slab: Area of 1 bar is pi*((16/2)^2) = 201mm2 (they have approximated to 200mm2). The bars are every 175mm, therefore the number of bars in a nominal 1000mm wide strip of slab is 1000/175 = 5.714 (multiplying by 5.714 is the same as dividing by 0.175). The total steel you can allow in your nominal 1m wide strip of slab is therefore 5.714*201 = 1149mm2 (or 1142mm2 as he has approximated).
Note you end up with a non integer (5.714) number of bars, which is fine(indeed correct) in this case as you are considering a 1m wide subsection of a wider slab, 5.714 is just the avg number of bars per metre width. You will end up calculating the bending capacity as moment/metre width of slab.
Hope this helps!