An option's value depends on several variables at once, and the greeks are the sensitivities to each. They are the language in which an options position is described, because the raw position, a list of strikes and expiries, does not tell you what you are exposed to.
Delta is sensitivity to the underlying price: how much the option's value changes for a one-unit move. It is also, loosely, the equivalent futures position the option represents. Gamma is the rate at which delta itself changes, which is what makes an options position non-linear and what makes hedging it an ongoing activity rather than a one-off. Vega is sensitivity to implied volatility. Theta is the loss of value as time passes, which is the cost of owning optionality.
The relationships among them are the practical content. A long option position has positive gamma, which means its delta moves in your favour as the underlying moves, and negative theta, which means you pay for that every day. A short option position has the opposite: it earns time decay and pays for movement. That is the same asymmetry described in volatility, stated in position terms.
Gamma is what makes a large options position affect the underlying market. A dealer who is short gamma must buy as the market rises and sell as it falls to stay delta-neutral, which amplifies moves. A dealer who is long gamma does the opposite and dampens them. Concentrated option positioning near a strike can therefore genuinely change how the underlying behaves, which is one of the few mechanisms by which derivatives positioning has a defensible effect on spot.
The reason to learn the greeks anyway is that they explain observable market behaviour. Expiry-related pinning, the way volatility rises when a large position is being hedged, and the asymmetry of option seller returns are all gamma and vega stories, and they are visible in prices whether or not you hold a book.