How come small firms can't affect one another's prices, but large firms can?
What you are talking about here is why firms in some market structures have to respond when their competitors change prices while in some market structures they don't have to change when their competitors change.
An oligopoly is the only structure where one company's prices affect anothers (and oligopolies only have large companies). In this market structure, there are only a few firms competing for market share. If one company lowers its prices, it will gain a huge advantage over the others.
In monopolistic competition, there are too many firms for this to happen. Imagine Chinese restaurants in New York City. If one of them lowers its prices, it's not like people from all over the city are going to go to that restaurant and hurt all the other ones.
It is not the absolute size of the firm that determines its ability to affect the market price. It is the size of the form in terms of percentage of total market demand it is able and willing to supply. Also it should be noted that a firm can decide only its own prices. It cannot decide on the prices charged by other firms in the market. When a firm changes is prices the other firms may or may not change their prices. also the exact increase or decrease in prices will be decided by each firm independently.
When a firm is meeting only a small percentage of the total market demand, any any increase in market price will induce customers to shift to other suppliers, and the demand earlier met by the firm will be distributed to the its competitors, who are so numerous that the change in the sale of an individual competitor firm will be insignificant to induce them to change their price.
On the other hand, when the firm reduces its price below the market price, all the customer will want to shift their purchases to this firm. Thus theoretically the entire market demand will shift to this firm. However, as the firm has the ability and capability to meet only a small percentage of the total market demand, it will not be able to meet all this entire demand. It will only be able to increase its supplies marginally above the original supplies level. Again the impact of shift of demand on individual competitor will be too small to induce them to reduce their prices.
It is worthwhile to note that the price adjustment process described above exist only in markets that approach perfectly competitive markets. In oligopolistic markets, the prices of one company are likely to have significant impact on prices of other firms in the market. In monopolistic market there are no other firms to be affected by the prices of the monopoly firm.