Loss functions encode our objectives for the learning system. How is this encoding guaranteed? I understand that loss functions encode how we want our learning system to behave in a sense. For example, take a binary classification task.
There are two types of error: False negatives, and false positives.
Claim 1: If our error function penalizes false negatives more than false positives, then after learning our system will be more likely to make the error of a false positive.
Is this claim true?
And if so, how does this encoding happen exactly? I feel that it's an automatic consequence of minimizing the loss function but what's going on behind the scenes?

Yes, your claim is correct given that system converged to a good local minimum. As you mentioned, this is an automatic consequence of the loss function. The encoding is in the loss function. Behind the scenes, it's just the minimization procedure, e.g. gradient descent, that makes it real.
For example, although probably not the best option, a loss function of the following form (let $\hat y$ be a sigmoid output) will enable some basic mechanism to choose between the two:
$$L(y,\hat y) = \begin{cases}y > \hat y && \lambda (y-\hat y)^2\\ y \leq \hat y&& (y-\hat y)^2\end{cases}$$
If $\lambda>1$, the error will be larger when $y=1$, so the optimization will try to make as little error as possible when $y=1$, e.g. trying to decrease false negatives, this can result in making more false positives. For example, if $\lambda\rightarrow \infty$, the best option is to choose $\hat y=1$ always, making a lot of false positive errors.
