Origins
To invade is simply to walk in — with hostile intent. The word comes from Latin invādere, from in- ('into') and vādere ('to go, to rush, to advance'). Roman generals did not need a special word for conquest; they just described the movement.
Latin vādere was not gentle. It described purposeful, forceful movement — a march rather than a stroll. Combined with in- ('into'), it produced the military term for entering enemy territory. The word carried its Roman military associations intact through French and into English.
The same root created a neat trio of directional words. Invade: to go into. Evade: to go out of, to escape. Pervade: to go through completely. Each preserves the core meaning of movement while adding a Latin prefix that specifies the direction.
Proto-Indo-European Roots
English wade — to walk through water — may share the same ancient PIE root *wadh- ('to go'). If so, there is a satisfying symmetry: wading through a river and invading a country are, at their deepest etymological layer, the same act of pushing forward through resistance.
The word invasion arrived separately, from Latin invasiōnem, the noun form. By the 15th century both verb and noun were established in English, permanently associated with military aggression.