Induced drag is a consequence of producing lift. Lifting line theory defines this in terms of circulation. But in simple terms, if you are producing lift, then you are producing downwash, you can't escape from that.
When looking at an aerofoil section, the apparent angle of incidence is increased by the downwash, and so a tiliting of the lift vector backwards.
So what's all this about vorticies then? At the tip of an aerofoil, the air rolls round from higher to lower pressure (in simple terms). This produces an increase in the local downwash towards the tip adding to the induced drag effect. In a low aspect ratio wing, this is proportianally much higher than in a high aspect ratio wing to the total lift/drag. When not interested in optimising for compressibility effects etc. the aim is to produce a uniform downwash across the span. This represents the minimum induced drag any given aerofoil can achieve.