A l'issue de l'audition, le député de Haute-Savoie expliquera qu'il était....

00:55 Mar 27, 2016 

Whatever you do in life, don’t asume the use of tenses in French and English is the same. It ain’t.

What’s happening here, in (very) simple/istic terms, is that the expresión “a l’issue de l’audition” precisely defines a moment in time with effect from which everything else in the sentence is expressed in the future tense.

12:35 Mar 26, 2016 

Your gut feeling is correct. This is a common use of the future in French, which has several options for a rendering in English, depending on context. Your instinct tells you that the past tense is needed here. Unless there is a stylistic reason to use another tense (and others are possible), then the past tense works perfectly well. Writeaway's suggestion of "went on to explain" is another perfectly good solution.

If you are not familiar with this type of use and/or need a quick reminder or gen up, this may help : http://www.french-linguistics.co.uk/grammar/future_tense_usa...

To see if the meaning of "to go on to + infinitive" fits, a sort of emphasis on a sequence of events, you might bear in mind whether "à l'issue de" is sufficient to justify that emphasis. There may be something in the text leading up to this point, in which case you may find it justified.
Check EN sources for meaning if you think it useful for your translation.
See : http://dictionary.cambridge.org/dictionary/english/go-on

12:13 Mar 26, 2016 

explained or went on to explain. just a choice of style. French has loads of options.