Gödel was fascinated by Leibniz's ideas, to the point that others felt he was obsessed: he checked out every book on Leibniz from his university library. He believed (correctly, I would say) that Leibniz's most important ideas (the characteristica universalis) had been nearly forgotten by society; but he also believed that this was due to a shadowy conspiracy meant to prevent the intellectual advancement of mankind. While one could make up a marvelous conspiracy theory about this, involving Newton, the Illuminati, the Invisible College, and so forth, it was more likely due to the fact that many of Leibniz's writings have never been published, and that Leibniz himself never completed the project.
At any rate, Gödel wanted to achieve Leibniz's dream of an exact, computational philosophy, able to come to provable conclusions. Gödel wrote, "There are systematic methods for the solution of all problems (also art, etc.)" Leibniz believed that the natural world arose out of a network of binary relations. This idea of a mathematical world underlying the world we see, a kind of Platonic realism, was appealing to Gödel as well, and he saw his work as pointing in that direction. Gödel like Leibniz, believed that the study of mathematics could tell us ultimate truths about the nature of reality. Since, as he proved, it is impossible to prove certain true facts about the mathematical universe, those truths must exist, he thought, somewhere outside of proof.
Gödel's more mathematical ideas were very important to people like Alan Turing, Stanislaw Ulam, and John Von Neumann. Gödel's famous proof of the incompleteness theorem needs to be able to make statements about mathematics using mathematics itself, and this required the invention of something very much like a programming language. Turing's key paper "On Computable Numbers, with an Application to the Entscheidungsproblem" uses the incompleteness theorem, proved five years before, to prove that it is impossible to decide algorithmically whether a given Turing machine will ever halt.
There are two main ideas I tried to get across in Machinamenta, that I laid out in the introduction. One is the idea of the kaleidoscope pattern, which I'm not going to go into here. The other is that the history of computers is not just the history of the development of mechanical math machines. There has also been, for a long, long time, a desire to make machines that can take ideas, and combine them with other ideas, to come up with new ideas. You see this in divination machines, which inspired Ramon Llull. Lllull's own devices inspired Leibniz to develop a much more ambitious and realistic plan. This in turn was taken up by Babbage and Gödel, who were direct influences on the people who built the first electronic computers. The dream of machine intelligence was already fully present through this chain of influence at the birth of the computer.