In the reinforcement learning setting, an agent interacts with an environment in (discrete) time steps, which are incremented after the agent takes an action, receives a reward and the "system" (the environment and the agent) moves to a new state.
More precisely, at time step $t=0$ (the first time step), the environment (including the agent) is in some state $s_t = s_0$, takes an action $a_t = a_0$ and receives and reward $r_t = r_0$ and the environment (including the agent) moves to a next state $s_{t+1} = s_{0 + 1} = s_1$, which will also be the state that the environment will be in at the next time step, $t+1$, hence the notation $s_{t+1}$. Here, the subscripts $_t$ refer to the time step associated with those "entities" (state, action and rewards). So, after one time step (or after $t=0$), the agent will be in state $s_{t+1}$ and the new time step will be $t + 1 = 0 + 1 = 1$. So, we are now at time step $t=1$ (because we have just incremented the time step) and the agent is in state $s_{t} = s_1$. The previously described interaction then repeats: the agent takes an action $a_{t} = a_1$, gets the reward $r_t = r_1$ and the environment moves to the state $s_{t+1} = s_{1+1} = s_{2}$, and so on.
In your summation, we are just discounting the rewards using a value denoted by $\gamma$ (which is usually between $0$ and $1$), that is often called the "discount factor". That summation represents the summation of the rewards the agent will received starting (in this case) from time step $t=1$. We could also just have $r_1 + r_2 + r_3 + \dots $, but, for technical or mathematical reasons, we often "discount" the rewards, that is, we multiply them by $\gamma$ (raised to a power associated with the time step that reward will be received).
In the above description, I said that, at some time step $t$, the agent takes an action $a_t$ and receives a reward $r_t$. However, it is often the case that the reward received after taken an action at time step $t$ is denoted by $r_{t+1}$. I think this is a little confusing, but not conceptually "wrong", because one might think that the reward for having performed an action at time step $t$ is only received at the next time step. (You should get used to slightly different notations and terminology. At the beginning, it is not easy to understand, if the notation is not precise and consistent across sources, but you will get used to it, the more you learn about the topic, in the same way that you get used to a new language).