I disagree fairly strongly with Daniel (to a point). To be a capable with the programming languages, api's and tools that you use, it strong math skills aren't necessarily required. However, if your goal is to go beyond the level of coding to other people's specifications being able to apply math and more complex algorithms to a problem (I consider these to be very similar) will take you a long way toward that goal.

Now for the why. In my experience, it's usually not hard to come up with programming solutions to many propblems that meet the requirement of producing the required output and keeping to a few restrictions. Where the math will really help you is when you have to take an existing solution and make drastic improvements to it. For example, if you need to write you own text comparison code, it isn't hard to use the search text to generate a simple regex to account for some spelling differences, but if you want to be able to emulate the behavior of Google Suggest, you need to understand algorithms like Soundex, Levenstein Distance and others. For a more mathamatical problem, consider a cache for keeping commonly used objects in memory instead of hitting the DB every time. How can you generate hash values in such a way that two objects with trivial differences have the same hash value?

At the end of the day though, the type and amount of math you need really depends on the field you're in. If you working with computer graphics, strong calculus and algebra skills are pretty important, but developers of financial software can probably get manage without. On the other hand, in any application where you have to deal with floating point arithmetic, having a strong understanding of number theory is a necessity, otherwise you'll constantly be faced with running totals that no longer add up correctly (due to rounding, truncation, and number representation issues).