After an RNA has been transcribed, in eukaryotes it is spliced before it leaves the nucleus. This means that parts of the RNA are removed (called introns) and the ends are capped. The parts left over in the mature mRNA after removing the introns are called exons.
The mRNA does not have to start with a start codon. There can be sequences before and after the bit which actually gets translated.
Yes, only one strand of DNA is transcribed into RNA. When you look at the mature mRNA below, read the first few bases and try to find their complements in the original DNA because this is where they were transcribed from. UCAUG is transcribed from the DNA AGTAC (with TCATG on the opposite side).
Now you look for the end of the mRNA. Bear in mind that all the A's are the poly-A tail which is added during splicing and determines how long the mRNA will persist in the cytoplasm. So you look for CUAGG in the original DNA, which must be transcribed from GATCC (with CTAGG on the opposite strand).
As you can see, that's exactly where the two red boxes start and end.
Now since the mRNA you see here is mature (indicated by the poly-A tail and the 5'-cap), that means introns have already been taken out. So any bases that you can see in the DNA between the start and end that we just found but not in the mRNA must have been an intron. Or the other way round: all the bases that are in the mRNA which you can also find in the DNA must be exons.
You will notice that exactly the bit that isn't in the red boxes doesn't appear in the mRNA anymore. Or: All of the mRNA is in the red boxes. The sequence is interrupted by a short bit which you can't find in the mRNA anymore - so this must be an intron.