Can that question be answered in a way that could be communicated to a friend over a beer at the local pub?

The answer to the second question is , no.

After studying and researching this topic for about 2 weeks, I have come to the conclusion that it cannot be communicated to outsiders. It is simply out of the reach of laymen. It may even be out of reach for most college graduates. Only grad students in mathematics might be able to "follow" the reasoning.

The best I can do for you is to try to give a rough overview of what the physicists were doing when they came up with this magical number 10 for the dimensions of spacetime. (For totally unrelated reasons) some physicists were already looking at strings to act as a replacement for point-particles in Feynman diagrams. The various masses of the fundamental particles came to be associated with tension in these strings. The tension would be expected to be higher when the string is in a more "excited" state of oscillation. Many quantum field theories are consistent at lower energies (/masses) but become inconsistent at higher energies. These cut-off points are called ultraviolet divergences (variously UV divergences, UV cutoff points, etc).

To avoid the UV divergences , and still have a consistent theory, physicists were forced to find something called the critical dimension. The 'D' here would be the full 10 dimensions, and the critical dimension is (technically speaking) D-2. In the diagram for a string moving through time (sweeping out a sheet), we are interested in the dimensions in which the string oscillates such that those dimensions are transverse to the direction of motion. (Thus D-2 is what we want, not D directly.)

The first thing physicists did in the late 1970s early 1980s was to find the critical dimension (D-2) for bosons. The magic number turns out to be 24, thus we add 2 to retrieve D, and get 26 dimensions for spacetime.

Later, they tried to integrate fermions into the same framework, and the "Critical dimension" ironically went down rather than up. This time the magic number came out as 8. Adding 2 to retrieve D, we get 10.

In both cases, the physicists were trying to find the "ground state" of the field, and then characterize the total number of dimensions that a harmonic oscillator must be oscillating within. Actual physical particles themselves would then constitute higher energy oscillations into those extra dimensions.

This is the simplest ("simplest") article on this topic that I could find on wikipedia. Notice it contains no equations.

https://en.wikipedia.org/wiki/Conformal_anomaly

a quote from that article :

In string theory, conformal symmetry on the worldsheet is a local Weyl symmetry and the anomaly must therefore cancel if the theory is to be consistent. The required cancellation implies that the spacetime dimensionality must be equal to the critical dimension which is either 26 in the case of bosonic string theory or 10 in the case of superstring theory. This case is called critical string theory. There are alternative approaches known as non-critical string theory in which the space-time dimensions can be less than 26 for the bosonic theory or less than 10 for the superstring i.e. the four-dimensional case is plausible within this context. However, some intuitive postulates like flat space being a valid background, need to be given up.

The more masochistic readers might consult the following article. The article was humorously named :

"A simple and rigorous proof of 26/10 dimensions in string theory …does not exist."

https://rantonels.github.io/critical-dimension/