In Mathematical finance, the concept of "arbitrage-free" portfolios is usually introduced before risk neutral measure. No arbitrage theory is intuitive for student when formulated this way: "there is no strategy that can beat the market". (or NFLVR)
Then one can show that this implies the existence of a risk neutral measure (completeness of the market gives uniqueness of the market). Finally the reverse implication is mathematically showed, but has little less interest because one can grasp the idea with the first proof.
Why is this equivalent ? The value of an asset is not his expected value at the end of the contract because there is risk. No Artbitrage hypothesis implies there is a common value for bearing the risk. If you mathematically change your probability space to take this common risk bearing premium into account in your metric, the value of the asset will be the expected value in the new probability space. This is mainly a mathematical trick, it's just factoring the risk premium.
Why is this interesting ? It's easier to find the risk neutral probability once and for all assets and then pricing the assets taking their expected values than calculating each asset expected value corrected by the risk premium.