What it actually means:
1. Let's assume that you sued someone for defamation of character, based upon a public statement that they had made,implying that you had a drink problem. If they were aware of the fixed penalty notice they could refer to it in court as evidence supporting the (alleged) truth of their statement. Similarly if a shopkeeper sued you for damaging their window, while allegedly drunk, (irrespective of whether any criminal action was involved), the shopkeeper could ask the court to accept evidence of the fact that you've been intoxicated in a public place (i.e. the FPN) as part of his case against you. Both of those are examples of the FPN potentially carrying weight in civil proceedings (which is what the last part of your quote refers to).
2. Let's now assume that, some time after you'd accepted the FPN, CCTV evidence came to light, showing someone fitting your description breaking that shopkeeper's window. If a criminal case was brought against you and you tried telling the court that you were at home in bed, totally sober, at the time of the offence, the FPN could be used as evidence to show that you were found intoxicated, 200 metres away from the broken window, less than 10 minutes after it had been broken. That's an example of 'taken into consideration in any subsequent criminal proceedings'.
3. During trial proceedings, a CPS barrister is normally prohibited from making any reference to a defendant's previous convictions/cautions/FPNs. However he can ask the court to admit such evidence where it indicates a 'propensity' for the defendant to commit the type of offence with which he is charged. Such applications most commonly occur with sexual offences. (e.g. "the defendant has previously been convicted of sexual assaults where he tied the victims' hands with their own tights, exactly as in the case now before the court"). It would be unusual for an application for 'propensity' evidence to be made (or granted) solely when such evidence only showed a single incident of public drunkenness. The only circumstances (which I can think of) where the CPS might seek to introduce such evidence in your case is where
(a) it was part of the CPS case that you were drunk at the time ; AND
(b) you tried to produce evidence to show that you NEVER got drunk.
Otherwise I can't envisage a court granting an application for 'propensity' evidence based upon a single FPN.