进 (traditional 進) carries the walking radical 辶 — to move forward, to advance, to go IN. (The traditional form had a bird 隹, which only flies forward, never backward — a perfect image of 'advance'.)
进 is 'to enter / advance': 进来 (come in, with 来 — 请进! 'Come in!' is what a host says at the door), 进去 (go in, with 去), 进步 (to make progress). It's the natural partner of 出 (out, next): 进出 = 'in and out'. From walking forward came every kind of advance — entering a room, improving a skill, moving up in life. 请进 is one of the first welcoming words you'll hear in a Chinese home.
Walking (辶) forward — to ENTER, to ADVANCE.
请进 ('please enter') is the host's welcome at the door. The traditional form 進 held a bird (隹) — birds fly only forward, never back, the perfect picture of 'advance'.