Trying to get the remainder of the filled grids (dataset 1) out of the bold polygon (dataset 2) . Would really appreciate some help.
First- is this raster or vector? The process is basically the same but you would use different tools for each:
The appears to be some value assigned to the colored squares so if you can simply make a selection base on that value not<null> or <> ' ' or what ever your empty value is, you would be able to either delete them from the original data or write them to a new feature.
I believe you could use the Select By Location tool in the Selection drop down menu.
Select features from <dataset 1> that intersect the source layer feature <dataset 2>. Then, as Joe indicates, you can delete them, or write them out to a new feature.
Retrieving data ...