The condition is mostly referred to as the No-Ponzi (-scheme) [NP] condition. It is one additional constraint, that prevents Ponzi-schemes: Paying debt with new higher debt, ad infinitum.
By the way: The NP condition is one condition, hence the associated multiplier should be $\psi$ instead of $\psi_t$. While certainly nothing is lost repeating the same condition over and over again (for any $t$), we don't need it more than once, and it is being imprecise.
Think about optimization for finite $T$ periods. Then, you have the condition that $B_T \geq 0$. The Lagrangian optimization gives you the local optimization between $0, 1, 2$... There are many solutions that are locally optimal, but you will only allow solutions that in the end lead to $B_T > 0$.
A simple example
Your example is much too messy to think about these core issues. Look instead at the problem
$$ \max_{\{c_t, a_{t+1}\}_t} \sum_t \beta^t U(c_t) + \lambda_t (a_{t+1} + c_t - Ra_t)$$
That is, a household that choses assets $a$ and consumption $c$ to maximize his utility. You can summarize the FOC as
$$ \beta^t U'(c_t) = \lambda_t \\
\lambda_t = R\lambda_{t+1}\\
\Leftrightarrow U'(c_t) = \beta R U'(c_{t+1})
$$
Look for a moment at the special case where $\beta R = 1$ (what does that imply?). With most preferences, this necessarily leads to $c_t = c_{t+1}$. This is the local optimization that I was referring to, which is what the Lagrangian gives you. There are, however, infinitely many solutions that satisfy $c_t = c_{t+1}$. Next, we try to use the budget constraint:
$$ a_{t+1} + c_t = R a_t\\
\Leftrightarrow R a_0 = \lim_{T\to\infty}\sum_{t=0}^T \frac{c_t}{R^t} + \frac{a_{T+1}}{R^T}$$
This is as far we get using the (infinite) set of local budget constraints, where I have used forward iteration (hopefully correctly), assuming any start date $t=0$.
Now, if the household also has to satisfy the NP condition, this boils down to
$$R a_0 = \lim_{T\to\infty}\sum_{t=0}^T \frac{c_t}{R^t}$$
which, as we showed $c_t$ to be constant, we can solve easily and receive a single budget constraint. The unique solution to the problem that satisfies the NP condition is the solution where $c_t$ is a constant and this last equation holds.
