Strictly speaking, everything you really need to know is contained in one line on page 31 of Eberron Campaign Setting:
[Infusions] function just like spells and follow all the rules for spells.
That effectively means they’re just a third type of spell along with arcane spells and divine spells, despite the initial “An artificer is not a spellcaster” statement at the beginning of that section. Since infusions follow all the rules for spells, that includes things like when a prestige class says it advances your spells per day or spells known: since those affects spells, they affect infusions, and those prestige classes can advance artificer.
But to have it truly spelled out for you (or a doubting DM), look to Magic of Eberron page 53, on which you’ll find a sidebar
Artificers and Prestige Classes
Artificers benefit in a specific way from prestige classes that have “+1 level of existing spellcasting class” as a level advancement benefit. [...]
This sidebar covers both how artificers qualify for prestige classes, and how they benefit from prestige classes, but the long and short of it is that you treat infusions just like they were spells, even though they technically aren’t. The one caveat is that they are neither arcane nor divine, so that does limit you somewhat.
Also keep in mind that the artificer does get pretty potent non-infusion features, including fresh craft reserve each level. When leveling in a prestige class, you give those up.