TLDR; a sharp or flat before numeral mean raise or lower the chord root from its normal diatonic spelling In the usual Roman numeral analysis we have these conventions:
First we have to give a label for a key like
E: for E major or Em: for E minor.
Then we have these points... capital numeral means major triad lower case numeral means minor triad 'o' after numeral means diminished triad '7' means a diatonic seventh above root a sharp or flat
before numeral mean raise or lower the chord root from its normal diatonic spelling That last rule is the crux of your question.
In E major the diatonic chord of the sixth degree is C# minor and gets labelled vi .
In E minor the diatonic chord of the sixth degree is C major and gets labelled VI .
If the VI from E minor is used in E major the chord root is altered from C# to C natural.
(This is called chord borrowing.)
When in E major that change in the chord root is made explicit in the Roman numeral figure by using the flat to show the change of chord root: bVI .
The reason your book's example is confusing is because the key signature on the staff is E minor, but the analysis key shows E which is read as
E major (it should be
Em: to mean minor)
and then the Roman numeral figures are labeled with flats which makes things seem like the assumed key is E major.
Basically the labeling is a mess and it makes confusing the use of sharps and flats on Roman numeral figures.
As others point out, the book doesn't even label the chords as 7th chords!
The book seems sloppy. @Tim makes a good point that Roman numeral analysis can be tricky with rock music.
The real conundrum to me is the final chord.
It's an E dominant 7th with a G natural on top.
That is a very bluesy sound.
It's a very familiar sound, but Roman numerals are a poor way to label it.
There is no way to indicate the simultaneous G# and G natural... and the minor 7th.
About the only thing the I label gets right is a root on E and a perfect fifth of B .
Tread carefully with harmonic analysis of rock music.
You might find a lot of inappropriate labels.
music.stackexchange.com/users/23919/Michael Curtis
The notation seems a bit confusing.
The first chord is a D7 in the key of either E-Major or e-minor; the author isn't quite so clear.
The key signature indicates e-minor in which case the chord is VII (no flat needed.)
If the author shows keys in ALL CAPS (which has been outdated since 1834), the chord VII7 but that disagrees with the single sharp signature.
The author also leaves off the seventh indicator.
Perhaps we can assume the author uses E irrespective of Major or minor mode and that he leaves off the seventh.
This still leaved the bVII vs VII
so it's still still.
The (V) must mean the seventh of G. The author shows a descending scale (the caratted numerals above some of the chords)
so perhaps that may help.
I think he's using the Roman numerals associated with E-Major but the score is written as e-minor; he's assuming that all chords have sevenths (OK, but limiting).
Then we have (if the score were in e-minor):
VII7, VII7, VII7, VI7, III, VII7, VI7, V7, IM7 (at least as I would notate it.)
music.stackexchange.com/users/23838/ttw
Are you confused because the song's in E minor, but the Roman numeral chord names are relative to E MAJOR?
Yes, that's how it works.
The numbers only care that E is the tonic.
E minor triad is i, E major triad is I.
The uppercase Roman number doesn't mean 'diatonic', 'in key' or 'correct', it just means 'major'.
Similarly for modified numerals.
G is the 'correct' third note of E minor, but G B D is called &flat;III.
That's &flat; because G is the flattened 3rd of E major, III because it;s a major triad.
G B&flat; D would be &flat;iii.
That's &flat; because it's built on the flat 3rd of E major, iii because it's a minor triad.
I repeat, these names would be the same whether we were in E major, E minor or E anything-else.
music.stackexchange.com/users/17003/Laurence Payne
RNA isn't designed for analysing 'rock' music.
It sort of works when non-diatonic chords need naming, as we use 'b' in certain circumstances.
As in key E. There is no D and no C, only D# and C#.
So, we have to call the D chord bVII and C chord bVI.
It gets as close as we can.
The bIII represents G, again because there's no G in key E:
G# is iii (III) but we need the flattened G#, making G. Thus bIII.
music.stackexchange.com/users/2808/Tim
The trouble is: The key sign is G-major/e-minor:
(in e-minor the flat should be a natural sign to explain the VI and VII degree are not of the harmonic scale!)
Indeed this progression is notated in respect of E-major!
D - D - D - C - G - C - D - E
If you transpose this figure to C you will understand that the flats are concerning Bb - Bb - Bb - Ab - Eb - Ab - Bb - G - C (with the blue note b10)
music.stackexchange.com/users/56219/Albrecht Hügli
I am working my way through an article on transformations in rock harmony, but I am getting stuck trying to read the author's roman numeral analysis.  I would really appreciate some help understanding how to read flattened roman numerals. Here is an example of what I mean:   Thank you!
Trouble reading roman numeral notation with flats
music.stackexchange.com
<roman-numerals><functional-harmony>
music.stackexchange.com/questions/81665
music.stackexchange.com/users/51866/286642
RNA isn't designed for analysing 'rock' music. It sort of works when non-diatonic chords need naming, as we use 'b' in certain circumstances. As in key E. There is no D and no C, only D# and C#. So, we have to call the D chord bVII and C chord bVI. The bIII represents G, again because there's no G in key E. G# is iii (III) but we need the flattened G#, making G. Thus bIII. RNA isn't designed for analysing 'rock' music. 
 RNA isn't designed for analysing 'rock' music. It sort of works when non-diatonic chords need naming. There is no D and no C, only D# and C#. The bIII represents G, again because there's no G in key E.