It works exactly the same as with any other multiclass character. In your case, your Sorcerer level will increase to 6 and you will gain all the features that a Sorcerer would normally gain when advancing to that level.
The advanced spellcasting ability of the prestige class (in this case Dragon Disciple) increases your effective Sorcerer level for the purpose of determining your spells per day (and, because a Sorcerer is a spontaneous caster, your spells known). It is simply a number that you add to your actual arcane spellcasting class level (in your case Sorcerer) to determine your effective level.
From the d20PFSRD for the Dragon Disciple, the FAQ/Errata box mentions:
The increase to his spellcasting level does not grant any other benefits, except for spells per day, spells known (for spontaneous casters), and an increase to his overall caster level.
In your particular case, as a Sorcerer 5/Dragon Disciple 8 you will have an effective Sorcerer caster level of 11 (5 from your levels as a Sorcerer and +6 from the Dragon Disciple's Spells per Day class feature).
At this point you will have the following Spells per Day and Known Spells:
lvl 1 2 3 4 5
per Day 11th 6 6 6 6 4
Known 11th 5 5 4 3 2
When you advance to Sorcerer 6 you gain all the benefits that a level 6 Sorcerer would normally receive, however, because of your Dragon Disciple levels your effective level for the purposes of Spells per Day and Known Spells will be 12.
lvl 1 2 3 4 5 6
per Day 12th 6 6 6 6 5 3
Known 12th 5 5 4 3 2 1