"A few cells in your body are about a meter long. These are the neurons running from your spinal cord to your feet." This statement, as you'll have read while looking at the problem, is in relation to the idea that the body has developed an efficient transport system to move proteins, organelles etc to the peripheries of the body, over the conventional diffusion process, so as to create a more efficient system. If we were to think about the process of diffusion across a meter, then we certainly know that it would take a rather long time.
e.g. if we assume the diffusion down the leg lasted for 1m exactly, and the viscosity for the sake of this equation was the same as water (=1*10-3), and assume 1-dimensional motion, then it would take 500 seconds, which when you compare it to how fast axons can act, it really is a massive difference.
Having started on this point though, it would also be good to provide some other examples. The one I'll mention is the movement of oxygen around the body. There are some insects and other small animals which actually move oxygen etc around their body through the use of diffusion. However, we as humans have a much more advanced method of moving air around, in which we have defined blood channels (vessels, arteries, capillaries) which can move the oxygen around in a more quick, efficient way. Again, if you look at the problem discussion I made above, it took 500 seconds to move something from the top of the leg to the bottom, so imagine if we were trying to move something around the whole body, especially when it is something as crucial as oxygen.