Consider reading "Reinforcement Learning, an Introduction" by Rich Sutton[0]. It is very accessible (and probably the most used textbook in the field), and goes into detail about the connection between RL and MDPs.

If you want a high-level answer right now about how the two are related: reinforcement learning focuses on predicting or maximizing the discounted sum of rewards, G_t = R_{t+1} + γ R_{t+2} + γ^2 R_{t+3} + ... which is called the return Crucially, the process generating the rewards (the environment) is assumed to be memoryless[1]. So we can reformulate the return as a recursive equation: G_{t} = R_{t+1} + γ G_{t+1}, since the return from "t+1" doesn't depend on the reward you got from the previous time step ("R_{t+1}").

This in turn allows you to define Bellman equations (which define optimal solutions to the problem of maximizing return, which is what you want to do). More generally assuming the problem is Markovian makes things substantially easier to reason about, because you don't have to worry about complex histories leading up to the agent's arrival in a state.

While it's honestly a pretty strong assumption, it's one that we make across many fields and particularly in RL. It's actually not too inaccurate; for example most games are Markovian (or close), and in physical problems (e.g. robot locomotion) if you've got position+velocity then controlling the robot can be viewed as a (continuous-time) MDP.

The weaknesses are obvious: not every environment is Markovian, sometimes there is long-term dependence on the past. For example, translating prose or poetry in such a way that preserves the point across is very tough[2], not really something you can formulate as an MDP. Other times, the weaknesses are easier to overcome (think of a poker game, where you can combine the present state information with the betting patterns from the past; this augmented state should be able to tell you everything you need to know).

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0. http://incompleteideas.net/book/bookdraft2017nov5.pdf

1. Another way of phrasing this is that each state provides all the useful information, and knowing about previous states does not tell you anything new about the environment. Think of perfect information games, like Chess. It doesn't really matter how you got into a particular position, just the moves you make from there.

2. Compare the poetry translations of Jerry Lettvin with what you'd get using existing translation software (https://sites.google.com/site/lettvingroup/Home/Projects-His...).