Every time I hear, "You should defrag your disk!" I roll my eyes and chuckle a bit.
There is absolutely nothing wrong with defragmenting your hard drive. Defragmenting reorders the data on the hard drive so that it is contiguous. "They say" that you will see a noticeable difference in speed, specifically faster. This was more true in the past, than it is now. Modern OSs, drives, and file systems benefit little from defragmentation. Yes, there are some cases where it makes a noticeable difference, but those are rare. Those occur on heavily used file systems where there are lots of writes/deletes going on. However, that is not the typical home or office user. A whole industry has popped up around defragmentation, and I personally feel its mostly snake oil. There are some benefits from advanced defragmenters, like moving data to the innermost track, etc. However, pure defragmention really provides little or no noticeable improvement.
Now as for your computers slowing down after a completed defrag... I honestly believe its not true. I think you are consciously looking for a difference in speed (which probably didnt happen) and therefore, subconsciously you think its slower.