Can anyone explain why the artificer spell levels only reaches level 5? Is it so that you can multi-class to get more expanded character? 

Because Artificers aren't intended to be full spellcasters like Wizards. They instead rely on their magical inventions and infusions to gain more power. You aren't intended to multiclass (though you certainly can if you'd like to).

Paladins and Rangers also only go up to level 5, for the same reason. These classes are usually referred to as 'half-casters'

OK thanks. You help answer another question as to what is a "half-caster"? Makes sense now.

For the record there are also Third-Casters (1/3) - the Arcane Trickster Rogue and Eldritch Knight Fighter. They only learn up to 4th level spells.

They are called Half and Third casters because that is their contribution to total spell slots when multiclassing spellcasters as per the multiclass spellcasting rules.

Spell Slots. You determine your available spell slots by adding together all your levels in the bard, cleric, druid, sorcerer, and wizard classes, half your levels (rounded down) in the paladin and ranger classes, and a third of your fighter or rogue levels (rounded down) if you have the Eldritch Knight or the Arcane Trickster feature. Use this total to determine your spell slots by consulting the Multiclass Spellcaster table.