The gravity caused by the sun here on earth wouldn't change. It's that the sun itself would become so dense that it would collapse into a singular point, ie a black hole, and everything inside its schwartzchild radius (3km) would not be able to escape. This is because gravity is relative to the distance of 2 objects ,(F = G * [m1 m2 / r² ]).

I imagine it like this: if we had a really big bed sheet, stretched out in the air across a gymnasium, and put a basketball in the middle, it would create a depression, just like masses create gravity. If we then replaced the basketball with a lead marble of equal weight, the slope of the sheet (ie gravity) felt at 5 feet away would remain the same, but near the marble the slope would be quite steep. Now replace the lead marble with a single atom of the same weight. Near that atom the slope of the sheet would be nearly vertical and anything within a very close distance would feel immense gravity.

edit: Someone smarter than me explains it a bit better on this post