Consider a swept wing airliner, and the direction of the crosswind - the upwind wing "sees" the airflow coming at it more directly, as if there is less sweep; and the downwind wing "sees" its sweep angle effectively increased.
Next have a look at the effect of wing sweep on the CL vs angle of attack graph. The straight wing has the steepest gradient, and more sweep reduces the gradient of the graph.
Finally, realise that as you rotate, you are increasing the angle of attack from something near 0 to something near 10 or more. So it is obvious that the upwind "less swept" wing will generate more lift unless you do something about it. Hence into wind aileron input is required.
On the B747 I fly, the aileron input required on the runway should roughly double as you rotate (steady crosswind).