Question About Kailath's Paper - An Innovations Approach to Least Squares Estimation Part I: Linear Filtering in Additive White Noise
They used to have different convention for writing this stuff back then.
But actually what you saw is really simple.  

It's all based on the Orthogonal Principle of MMSE.

They say in (9) that the Additive Noise is uncorrelated (By defining its Auto Correlation by Delat Function).
In (10) they say the optimal estimation of $ x \left( t \right) $ is given by $ \hat{x} \left( t \mid t \right) $ is a linear combination of $ y \left( t  \right) $ and $ z \left( t \right) $.
In (11) they exactly use the Orthogonal Principle which the estimator must obey.  

In (12) they just derive the Correlation Matrix between the processes.
Since the linear estimator represent the correlation it is not surprising to see that it is the linear combination defined by the optimal filter $ g \left( \cdot \right) $.

The trick here is the fact (Using article notations) $ \overline{ \left( x \left( t \right) - \hat{x} \left( t \mid t \right) \right) v' \left( s \right) } = 0 $  which means (By Linearity of the Expectation) $ \overline{ x \left( t \right) v' \left( s \right) } = \overline{ \hat{x} \left( t \mid t \right) v' \left( s \right) } $.
Expand the right hand term by (10) by plugging $ \hat{x} \left( t \mid t \right) $ and you get (12).

By the way, Kaliath has much better written, in my opinion, paper on the subject called A View of Three Decades of Linear Filtering Theory.