The DATE function creates a proper Excel date with given year, month, and day values. Then, it's simply a matter of comparing each date in the range with the date created with DATE. The reference B4 is fully relative, so will update as the rule is applied to each cell in the range, and any dates greater than 8/1/2015 will be highlighted.
Greater than or equal to, etc.
Of course, you can use all of the standard operators in this formula to adjust behavior as needed. For example, to highlight all dates greater than or equal to 8/1/2015, use:
Use another cell for input
There is no need to hard-code the date into the rule. To make a more flexible, interactive rule, use another cell like a variable in the formula. For example, if you want to use cell C2 as an input cell, name cell C2 "input", enter a date, and use this formula:
Then change the date in cell C2 to anything you like and the conditional formatting rule will respond instantly.