Someone asked me "I thought the school gave you funding? Don't you get a stipend? So why do you need to apply for separate funding?"
These are excellent questions which I did not thoroughly address. This academic year I am funded with a Teaching Assistantship. I have a 50% teaching load (20 hours per week). In exchange for being a TA, my tuition is waived and I get a monthly stipend. From my student stipendI still have to pay for my health insurance and some other miscellaneous fees (about $800 per semester I think). I have no loans for grad school and my undergraduate loans are deferred until I finish.
In biology, you shouldn't go to graduate school unless they're going to pay you. I think this is true for most Ph.D. programs in the sciences. Unless you come with your own funding, the source of your stipend will probably be a Teaching Assistantship, Research Assistantship, or perhaps a university fellowship. Schools and departments are limited to some extent in the number of TAships they can award. In some departments TAships are so scarce that you are basically required to come with your own funding (something like an NSF Graduate Research Fellowship which you can apply for before you being grad school).
So if I have funding this year, why do I need to apply for grants? There are two answers to this question. The first is that research is expensive. My advisor has some money, but not enough for me to do whatever I want. I need to apply for grants that allow me to use the money for supplies and equipment that I don't already have access to. Most importantly (for my research), I need money to cover my transportation costs and supplies for while I'm in the field. I'll most likely be doing my field work in Africa so simply getting there is a significant expense. Hopefully I can get a grant for a few thousand dollars this year to cover the expenses for an initial trip.
The second reason I should apply for additional funding is that some fellowships pay a lot more than being a TA. The NSF GRF pays you $30,000 per year for three years. That's about $8,000 more per year than a TA stipend. Somewhat strangely, the EPA STAR fellowship would actually only pay $20,000 per year for stipend, which is less than I make as a TA. The advantage (if you see it as one) of a big stipend-paying fellowship like that is that you don't have to teach so you should be able to progress more quickly in your research.
I hope this clarifies things somewhat. Please feel free to comment if you have other questions or have something else to add to the discussion from your own experience.