Gausian white noise and sequence space model

Question

Cosinder the stochastic Process (GWN):

$$ dY_t= f dt + \sigma dW_t $$

for W being a standard Brownian Motion and $f\in L^2([0,1])$.

It was said in the lecture that observing this process is equivalent to observing the sequence space model:

$$ y_k= \int_0^1 f\phi_k \,dt + \sigma \varepsilon_k $$ for $\phi_k$ being some orthotogonal basis of $L^2([0,1])$ and $\varepsilon_k$ being iid standard normal distributed.

My questions now are:

1. what does it exactly mean to observe the process $Y_t$?
2. does anyone know a proof for the equivalence? and in which sense are these models actually equivalent?