Unfortunately for your hurting head, there is not really a satisfactory answer to the question.
There is no way of measuring anything outside the visible universe so any answer would be pure speculation and beyond the remit of science.
When scientists refer to the universe they invariably mean the visible (or known) universe. The visible universe is EVERYTHING and EVERYWHERE we can observe.
The rate of expansion is determined by measuring how fast other galaxies are moving away from us using the doppler effect. According to Stephen Hawking this measurable effect shows that the universe is expanding - the galaxies are moving away from one another - by between 5 and 10 per cent every billion years. Rewinding backwards in time 13.7 billion years ago the distance between the galaxies would be zero.
So all we really know is that the universe that we can observe has got bigger. We cannot tell that it has got bigger "into" something.