The House turns the proposal into a bill, as required by the Constitution, editing it as it sees fit along the way. The executive can not create a bill. The bill then follows the normal constitutional procedure for introduction into the Senate, passage in both houses, reconciliation of the versions, and submittal to the executive for signature or veto.
This is the theory.
In actual practice, both House and Senate do separate budget bills at the same time, which are then reconciled.