Understanding subscripts in first order conditions of dynamic optimization problems
In an intertemporal maximization problem, we seek to find the optimal sequence  of the control and the state variables. It is the recursive nature of the problem that permits us to consider a "typical" point in time and just one condition per variable.  

For each such problem, we need to find out (carefully) in how many distinct periods a specific realization of a variable appears. To do this properly we should distinguish between the "absolute" index, and a "running" index. In the formulation of the Lagrangean as appears in the question, this is not done (and it is usual practice not to, but it may become confusing).  

So I would use the $t$ symbol as the absolute index (to arrive at same-looking first-order conditions), and some other symbol for the running index, say

$$\mathcal{L_t} = \sum_{j=0}^\infty \beta^{j}\left\{u(c_{t+j}) + \lambda_{t+j} \left[ f(k_{t+j}) + (1 - \delta)k_{t+j} - c_{t+j} - k_{t+j+1}\right]\right\} $$

Note that $t$ no longer affects  the discount factor $\beta$, and this is because the discount factor has to do with looking at the future, which is represented by the index $j$. Also, note that $j$ starts at zero, indicating that the first period is the $t$ period.  

Written this way, the Lagrangean says "we are at some point in time indicated by $t$ (that can take the value zero or whatever positive value), and we are looking forward period by period counted by index $j$".

For any given $j$ we have

$$\mathcal{L}_t =...+ \beta^{j}\Big\{u(c_{t+j}) + \lambda_{t+j} \left[ f(k_{t+j}) + (1 - \delta)k_{t+j} - c_{t+j} - k_{t+j+1}\right]\Big\} + \beta^{j+1}\Big\{u(c_{t+j+1}) + \lambda_{t+j+1} \left[ f(k_{t+j+1}) + (1 - \delta)k_{t+j+1} - c_{t+j+1} - k_{t+j+2}\right]\Big\} + ...$$

Pondering this, we realize that the variable $k_{t+j+1}$ will appear in only two consecutive periods, and so the first order condition for a "typical" element of the sequence $\{k_{t+j}\}_{j=0}^{\infty}$ can be exrpessed by differentiating only these two periods with respect to $k_{t+j+1}$. Doing so we get

$$\frac {\partial \mathcal{L}_t}{\partial k_{t+j+1}} =  -\beta^{j} \lambda_{t+j}  + \beta^{j+1}\Big\{ \lambda_{t+j+1} \left[ f'(k_{t+j+1}) + (1 - \delta)\right]\Big\} $$

Take common factors (which will simplify the discount factor) and set equal to zero

$$\frac {\partial \mathcal{L}_t}{\partial k_{t+j+1}} = \beta^{j} \Big[-\lambda_{t+j}  + \beta\Big\{ \lambda_{t+j+1} \left[ f'(k_{t+j+1}) + (1 - \delta)\right]\Big\}\Big] = 0$$

To lighten the indexing burden, we can express this for $j=0$, to obtain

$$\frac {\partial \mathcal{L}_t}{\partial k_{t+1}} = 0 \implies -\lambda_{t}  + \beta\Big\{ \lambda_{t+1} \left[ f'(k_{t+1}) + (1 - \delta)\right]\Big\} = 0$$