Use the angle addition formula for sine.

We can express $75^\circ$ as the sum of two standard angles, $45^\circ + 30^\circ$.
The addition formula for sine is $\sin(A+B) = \sin A \cos B + \cos A \sin B$.

We know that:
$\sin 45^\circ = \frac{\sqrt{2}}{2}$
$\cos 30^\circ = \frac{\sqrt{3}}{2}$
$\cos 45^\circ = \frac{\sqrt{2}}{2}$
$\sin 30^\circ = \frac{1}{2}$

The value of $\sin 75^\circ$ in simple surd form is $\boxed{\frac{\sqrt{6} + \sqrt{2}}{4}}$.