To be honest, terms like these are very confusing as they aren't clear cut and on one side of the border. They are more grayish.
I'm gonna tell you how I convinced myself, as I too had this confusion as soon as I read your question. But I managed to convince myself through this argument.
First of all we are gonna clear up 4 terms, Radiance, Irradiance, Differential radiance and Differential Irradiance.
"Radiance" is what you say associated with a certain direction. To be more formal and according to wikipedia,
It's the amount of radiant flux emitted/transmitted/received per unit projected area, per unit solid angle.
Next is differential radiance. We can think of it as an infinitesimal quantity of radiance emitted or recieved in a very small solid angle $d\omega$.
Next is Irradiance. Irradiance isn't normally associated with a direction. According to Wikipedia it's
Radiant flux received by a surface per unit area
But more commonly and what makes more sense to me, and to the answer of your question, think of irradiance as the integration of radiances over a set of directions.
So we can say
$E = \displaystyle\int_{\Omega} L(\omega)\; \omega.n \;d\omega\\\omega \in \Omega$
So if we integrate the radiances from every direction that leads us to the original definition of irradiance where direction isn't of concern. However usually we are concerned with only a subset of all the directions such as the Upper hemisphere or the lower hemisphere. This means for example,
$\Omega = \{ \omega : \omega.n \geq 0 \}$
As we can see here, we have limited the irradiance to a set of directions, the upper hemisphere. This doesn't necessarily change it into radiance which is associated by direction. Instead what this means is, when calculating irradiance we are concerned with the light coming only from these directions, although we haven't incorporated the directional quantity into the formula like with radiance.
This is the difference between irradiance from a certain direction and radiance. Think of it like this. You are holding a paper and there are 2 light bulbs in front of you. You want to measure the irradiance. Normally it would just be the radiant flux received by both bulbs per unit area. But now let's say I limit the direction so I am only concerned with the first bulb. Note that I am still calculating the "irradiance". If I move farther away the flux will decrease thus the irradiance even tho I am concerned with a specific direction. However this isn't the case with radiance where moving farther away won't change it since we divide by the solid angle too balancing the change.
The last quantity is differential irradiance. I thought of it as a tiny amount of irradiance from a specific direction. (Again direction gets involved)
If you think of irradiance as not assosciated with direction at all, even then when you try to think of differential irradiance you are gonna say it's a tiny amount of irradiance from a small range of direction or maybe a specific direction. That's the reason why it's small.
But if you think of irradiance as the sum of all the radiances over a specific set of directions. You'll see that it makes things clearer and you'll naturally arrive at the conclusion that differential irradiance will then refer to an irradiance from a specific direction.
So coming back to your question at last, I hope you might have gotten some intuition as to what "irradiance in a direction means".
Mathematically proving it is no hard feat tho. The answer here explains it quite well. I'm just gonna give a brief explanation. We know the rendering equation is given as
$L_{outgoing} = L_{emission} + \displaystyle\int_{\Omega} L_{incoming} \;f_{BRDF}(\omega_i, \omega_o)\; \omega_i.n \; d\omega_i$
Assuming for the moment that emission part is zero. we end up with,
$L_{outgoing} = \displaystyle\int_{\Omega} L_{incoming} \;f_{BRDF}(\omega_i, \omega_o)\; \omega_i.n \; d\omega_i$
Now as I wrote before, if you forget the BRDF for the time being, we are just integrating the radiances over a given set of direction which is the same as irradiance.
If we look at one instance of this summation/integration, it's gonna be
$dL_{outgoing} = L_{incoming} \; \omega_i.n \; d\omega_i \; f_{BRDF}(\omega_i, \omega_o)$
$dL_{outgoing} = dE \; f_{BRDF}(\omega_i, \omega_o)$
We put $d$ with the outgoind radiance and irradiance because it's a very small part (we are looking at just one instance of that summation/integration)
$f_{BRDF}(\omega_i, \omega_o) = \displaystyle\frac{dL_{outgoing}}{dE} $
Which is the ratio of the outgoing radiance to the incoming irradiance.
Again this was just the way I convinced myself and might have some mistakes. Though this is the best I came up with.
