Yes, thanks for the correction.
The key is selection. Random mutations are going to make random changes, but it is the selection step which is important. You can think of selection like a ratchet. Each time a beneficial mutation occurs the ratchet tightens down a click and moves the nut one rotation tighter. If a detrimental mutation occurs and pushes the ratchet the other way the ratchet spins freely and the nut it is attached to doesn't move because deleterious mutations are removed from the population by selection. This allows more and more beneficial mutations to build up over time while eliminating deleterious mutations.
The interesting aspect of all of this is the disconnect between DNA sequence and selection. What natural selection "sees" is function, it doesn't see DNA sequence. If there is a super complex interaction that produces a beneficial function, then it is selected for. If there is a super simple interaction that produces a beneficial function, then it is selected for. Overall, natural selection is blind to what is happening at the molecular level. All it sees is the ability of the organism to reproduce.
You might also be interested in genetic algorithms. These are used for engineering things like circuits or other physical structures. They use the same process, random changes followed by selection. This process often produces complex designs that even the humans running the experiments don't understand.
Dr. Adrian Thompson has exploited this device, in conjunction with the principles of evolution, to produce a prototype voice-recognition circuit that can distinguish between and respond to spoken commands using only 37 logic gates - a task that would have been considered impossible for any human engineer. He generated random bit strings of 0s and 1s and used them as configurations for the FPGA, selecting the fittest individuals from each generation, reproducing and randomly mutating them, swapping sections of their code and passing them on to another round of selection. His goal was to evolve a device that could at first discriminate between tones of different frequencies (1 and 10 kilohertz), then distinguish between the spoken words "go" and "stop".
This aim was achieved within 3000 generations, but the success was even greater than had been anticipated. The evolved system uses far fewer cells than anything a human engineer could have designed, and it does not even need the most critical component of human-built systems - a clock. How does it work? Thompson has no idea, though he has traced the input signal through a complex arrangement of feedback loops within the evolved circuit. In fact, out of the 37 logic gates the final product uses, five of them are not even connected to the rest of the circuit in any way - yet if their power supply is removed, the circuit stops working. It seems that evolution has exploited some subtle electromagnetic effect of these cells to come up with its solution, yet the exact workings of the complex and intricate evolved structure remain a mystery (Davidson 1997).
If you went back to the Cambrian you would also say that they looked complete. That seems to be a human bias.
Every species is an example of a transition between its ancestors and its descendants. All populations are evolving. As to stasis, if mutations don't confer a beneficial change then there is nothing to select for. As mentioned earlier, species can hit a local peak on the landscape of fitness which doesn't allow them to change much. In order to get to other peaks in the fitness landscape they would have to accumulate a lot of deleterious mutations to get to the valley below before they can climb another peak. Selection will not allow that to happen in some instances where the species remain in a relatively stable environment or niche.
The question I always ask is how one figures out that a fossil has no ancestors. We have only looked at a tiny, tiny, tiny portion of the fossil record. On top of that, only a tiny fraction of species were probably preserved, and only a tiny fraction of those fossils survive to the modern day in places where we can easily access them. Absence of evidence is not evidence of absence, as the old saying goes.
The argument at the time was that abiogenesis was impossible because biomolecules could not arise on their own. Miller disproved that single criticism. No one thinks that the abiotic production of amino acids proves abiogenesis, it is only one step in the process.
The fossil record is incomplete because the geologic record is incomplete, as is our search for fossils.
" For my part, following out Lyell's metaphor, I look at the natural geological record, as a history of the world imperfectly kept, and written in a changing dialect; of this history we possess the last volume alone, relating only to two or three countries. Of this volume, only here and there a short chapter has been preserved; and of each page, only here and there a few lines. Each word of the slowly-changing language, in which the history is supposed to be written, being more or less different in the interrupted succession of chapters, may represent the apparently abruptly changed forms of life, entombed in our consecutive, but widely separated formations. On this view, the difficulties above discussed are greatly diminished, or even disappear. "--Charles Darwin, "Origin of Species"
Darwin wrote an entire chapter dealing with the fossil record and how the geologic record is incomplete. You may want to give it a read. The incompleteness of the fossil record hasn't been a problem for the theory of evolution since its inception.
Scientists are people, too. The science itself is dispassionate, but it is argued over by very passionate scientists. If you ever attend a scientific conference you will see just how passionate (and stubborn, obstinate, pigheaded) scientists can be. If you have a fragile ego then the sciences may not be for you. The ethos in the sciences is that you beat the tar out of theories, and if they pass that test then they are probably good theories.
What the scientist is talking about in your post is the regret that creationists will misrepresent the science, which is a valid worry. Creationists are known for using quote mines and distorting what scientists say.