What is "academic freedom", when so many professors and post-docs do only the research that is being funded by external agencies?

While it is certainly true that the bulk of university research depends on external funding, and thus requires approval of the external funder, there are a number of ways in which a professor in the United States (and many other nations) can pursue research with almost complete freedom:

• Most professors own salaries are supported (at least at a "9 month" level) on the basis of teaching, not from research grants. A research position, however, often teaches only 1-2 classes per semester, however, leaving significant time for the professor to spend on research, whether or not that research also has external support.
• Some undergraduates are willing and able to work on research for course credit or even on a volunteer basis.
• Graduate students and postdocs are sometimes supported on fellowships that put no constraint on the subject of their research.
• Some institutions also allow professors to "bank" a portion of the research funds that they bring in, which may then be used later as unrestricted funds.
• Early career professors often have startup funds and later career professors often have endowed chairs, both of which typically have no constraints on the research they can be used to conduct.
• In extremely rare cases, a professor may get a "genius grant" or similar sort of external award that is a significant amount of unrestricted money.

The amount of resources available through such unrestricted routes is typically much smaller than if one can convince an external funder that one's work is worthy, but one can get quite a long way on preliminary work, proofs of concept, and small-scale experiments with rather small amounts of resources.

Combine this with the fact that there is a wide diversity of funders out there with different agendas and preferences, and one can see that academic freedom in research is very real indeed.

Typically, the prof wrote the project for which (s)he received the funding. This is how (s)he determines what to do research on. When a postdoc does not write her/his own funding proposal, then her/his freedom is severely limited.

Academic freedom isn't just about choosing what you work on (which was covered by the other answers). It also means that a researcher doesn't face consequences for what they research or for their conclusions, and the same for professors and what they teach (within the limits imposed by professional ethics, of course). In theory at least, if the conclusions of a study do not please whatever politician is effectively in charge (the minister for higher education or equivalent, or even above) or whatever private overseer, then the researcher/professor should not face consequences for this.

Think about for example an economics professor researching and teaching ideas that are not part of the orthodoxy, or an environmental scientist that reaches some conclusions regarding climate change that may displease an unfortunate fringe of politicians, or the same environmental scientist who finds that the actions of a powerful oil company (with friends in the government) are harmful, or a medical researcher who finds that some drug is inefficient / harmful, etc.

If lobbies had the power to influence politicians/private overseers who could fire professors and researchers at will, do you think there would be many studies who find that criticize the products/actions of big private companies? Of course, as you have noticed, funding for such research can be cut, which is a problem in itself.