Why are radians dimensionless?

There are quite a few related questions and answers here, here, and here. Despite that, I'm not voting to close as duplicate, because it seems like your question is different from the other three in that you're more asking what a "physical dimension" is. The third question is even answered with more or less the question you're asking. Related to this question is the answer here, which I think is excellent.

The long and short of it is that angles aren't intrinsically dimensionless. They're dimensionless by convention. I'll justify this in a moment.

Now. About "physical dimension." For a definition, see here.

However, I think "physical dimension" is a bad way to describe what dimensions of units are. It's more confusing than it needs to be.

Therefore I'll begin my answer by defining what units are formally and mathematically in the first place. Units are a way of assigning types to numbers. Units have their own arithmetic. One can multiply and divide units to get new compound units. For example $\textrm{m}/\textrm{s}=\textrm{m}/\textrm{s}$ is the type of a speed. Or $\textrm{m}\cdot\textrm{m}=\textrm{m}^2$. There's also the identity unit, $\textrm{id}$. This is the unit of a dimensionless quantity. For any unit $u$, $u/u=\textrm{id}$, or $\textrm{id}\cdot u = u$. Usually we drop the identity unit from numbers when we write down quantities "without units" e.g. we usually write $1 \textrm{ id}$ as just $1$, but you can think of there being an implicit unit there.

Now we can also define new units in terms of the old units. This is how most metric units are defined. For example $\textrm{N}:= \textrm{kg}\cdot\textrm{m}/\textrm{s}^2$. Now $\textrm{N}$ actually equals $\textrm{kg}\cdot\textrm{m}/\textrm{s}^2$, so the two are interchangeable, we just use $\textrm{N}$ as shorthand for the compound type. Similarly, it turns out that we define $\textrm{rad}:= \textrm{m}/\textrm{m}=\textrm{id}$. As for why we define it that way, many other answers on this page and the ones I've linked to answer that, so I won't address it in depth. Briefly, it's because we define the radian so that it measures a ratio of lengths. If $s$ is the length of an arc cut out by a central angle of radian measure $\theta$ of a circle of radius $r$, then $s=r\theta$, or $\theta = \frac{s}{r}$. Now some units also have a constant in front of the basic units in their definition. For example, since the arc length formula when $\theta$ is measured in degrees becomes $s=\frac{\pi}{180}r\theta$, $\textrm{deg} = 180/\pi \textrm{ id}$.

Now, on to the main point. In fact all SI units can be written in terms of seven base units. Why is the radian dimensionless even though it measures what could be thought of as a physical quantity or dimension?

Convention. As mentioned in this answer, one could add an 8th fundamental unit, say the radian. Then the formula for arc length would become $s= \frac{r\theta}{\textrm{rad}}$. All we need to do is introduce a new constant to the formula of dimension $\textrm{rad}^{-1}$, and everything would work just fine. The reason we don't is that we want to try to reduce the number of primitive units as far as possible. Relatedly, for historical reasons, the mole, which measures "amount of substance" basically just counts the number of atoms of that substance present, therefore it's sort of dimensionless. In fact there is a proposal to eliminate it as an SI base unit according to this answer here (maybe, it's a little unclear what the extent of the proposal is).

Essentially, you can take as the definition of the dimension of a unit that it's the way that unit is expressed in terms of the SI basic units (ignoring any constant in front). So $\textrm{rad}=\textrm{id}$ has dimension $\textrm{id}$, and $\textrm{deg}=\frac{180}{\pi}\textrm{ id}$ also has dimension $\textrm{id}$. Similarly $\textrm{ft}$ has dimension $\textrm{m}$, though we might express this in English in this case by saying that it has dimensions of length.

Now it might be a natural question to ask, why do we have all these other units then if they can all be expressed in terms of the basic units? What are they good for?

The answer is roughly three things.

1. Units keep track of extra semantic information. $1\textrm{ rad}$ is semantically different than $1\textrm{ id}$. Syntactically, they behave the same in the mathematics, however the first tells me that the quantity came from measuring some angle. Also $\textrm{N}/\textrm{m}^2$ is suggestive of pressure, compared to $\textrm{kg}\cdot \textrm{m}^{-1}\cdot\textrm{s}^{-2}$.
2. Brevity of course. Defining short names and symbols for commonly used compound units makes for shorter and more clear equations. (This sort of ties in with point 1).
3. To prevent us from doing something incoherent. It doesn't make any sense to try to add quantities with different units. $1\textrm{ m}+1\textrm{ s}$ makes absolutely no sense. This can be explained by saying that the dimensions don't match, i.e. that the expressions have different SI basic units. However, we also shouldn't be able to add $1\textrm{ rad}$ and (well the only other official SI dimensionless unit is the steradian, so I'll introduce the dimensionless unit $\textrm{item}=\textrm{id}$ which counts a quantity of something), $1\textrm{ item}$. I.e. $1\textrm{ rad}+1\textrm{ item}$ shouldn't make any sense. Thus having other units is helpful to prevent us from doing something that doesn't make sense with our quantities.

Edit:

Having written this post, and read the references I've cited, I'm now actually more in favor of the position of adding angle and count of stuff (as opposed to mole) as fundamental dimensions. But the whole point of my answer is that it's basically arbitrary and based on convention. In fact, I'd argue that the reason why one can't add $\textrm{item}$ and $\textrm{rad}$ even though both are defined to be equal to $\textrm{id}$ is that they ought to have different dimensions. On the other hand, as long as you convert to a common unit first, say $\textrm{rad}$, you can sensibly add degrees and radians.

Edit 2: for an answer along the same lines, I just noticed the answer here.

The circumference $C$ of a circle is related to its radius $r$ as $C=2\pi r$; hence, $C/r$ corresponds to an angle of $2\pi$ radians. But circumference and radius both have units of length, so their ratio is dimensionless. (To generalize this, consider a circular arc of radius $r$ and angle $\theta$ having arc length $s=r\theta$.) Hence a unit of angle must be dimensionless.