Slippage is often treated as a fee or a fault in the exchange. It is neither: it is the direct consequence of buying more than is available at the price you were quoted.
The quoted price applies to a limited quantity
The price displayed on any venue is the best available offer, and that offer exists for a specific quantity rather than for any amount you might want.
An order larger than that quantity is filled at the best price for as much as is available, then at the next price, and so on until it is complete.
The average across those fills is the price actually paid, and it is always worse than the first level touched.
Depth thins as you move away from the middle
Order books are densest close to the current price and become progressively sparser further out, because market makers concentrate their risk where it turns over fastest.
A large order therefore crosses widening gaps, and each additional unit costs more than the last rather than the same amount.
This is why slippage rises faster than order size. Doubling the quantity typically more than doubles the cost of execution.
Automated market makers price the same effect as a curve
A constant product pool has no order book, but the ratio of its reserves determines the price, and every unit bought shifts that ratio.
The result is a continuous version of the same phenomenon, with the price moving further as the trade grows relative to the pool's size.
Slippage tolerance settings exist because that final price cannot be known in advance, since other trades may execute in between.
Volatility widens the gap independently of size
Between placing an order and it being matched, the market continues to move, and in fast conditions that movement can exceed the effect of size entirely.
Market makers respond by widening quotes and reducing displayed depth, which makes execution worse for everyone during exactly the periods when people most want to trade.
Execution strategies trade time for price
Breaking a large order into smaller pieces over time lets depth replenish between them, which reduces the price impact of each individual fill.
The cost is exposure to price movement during the extended execution, so the choice is between a known impact now and an unknown drift later.
Limit orders avoid slippage entirely by specifying the worst acceptable price, at the cost of possibly not being filled at all.