I think there are several reasons.
The most important one in my opinion is that as a professor in an experimental field you really don't have a choice. In many fields, grants pay for all the lab personnel, experimental equipment and even the professor's own salary. This means that if a professor has no money, her lab is effectively closed and in some institutions she may be effectively fired (given no salary and no workspace) even if she has tenure. This has dramatic consequences for everyone in the lab, and once the lab is closed it won't help if the professor suddenly get new funding. It is somewhat like a CEO who has to make sure the company doesn't close.
This is a situation you need to avoid at all costs. Since grants are very competitive, you need to submit many grants to minimize the probability of having no grants at some point in time. Additionally, one grant will often not be enough to cover all the expenses.
So most professors in this situation really don't have a choice, they simply have to do it even if they want to spend most of their time on research. In my experience, most professors would clearly prefer to spend all of their time doing science rather than grant-writing and administrative stuff.
A second reason, is that there is that there is a clear correlation between funding and research productivity. For example, if you have more people in your lab then you have more projects and eventually more papers. Also, many high-impact journals like to publish large datasets (personally I think this is because they get a lot of citations) which are basically just a factor of the amount money one has. Additionally, academic institutions in the US take very high fractions of grant money as overhead (can be over 50%), so they strongly encourage their faculty to get grants. These factors all mean that getting grants can have a direct impact on research output, promotion and prestige.
Finally, you need to consider that in many fields the professor's main job is not executing research. A professor needs to teach, participate in committees, obtain funding, manage the lab administration, promote the lab's work, recruit personnel and supervise students. How then can she spend most of her time doing research? Imagine a software company where the CEO not only needs to run the company but also has to manage HR, PR, funding and all administration. Would you expect that CEO to spend most of her time programming? That should clarify why you don't see many professors as first authors (or CEOs as lead programmers).