For starters, in a tightly-knit community, everyone contributes to helping women through pregnancy, birth, raising the child, etc. Although parents may give preferential attention to their own children, this should still weaken the pressure on men to "fuck and run" (and, at the same time, the pressure for women to find a committed, monogamous partner).

If everyone contributes, and there is minimal preferential attention to their own children, then why would anyone do something besides fuck and run? Long-standing relationships come from male parental investment, across species.

Furthermore, as NancyLebovitz has already pointed out, close social ties make it easier to enforce sexual selection for more attentive and nurturing partners, since you've got a reputation to maintain.

It only matters whether your sexual partner is attentive and nurturing towards your children. Whether or not they're attentive and nurturing towards you only determines their value as friends.

The summary of Sex at Dawn that Nancy linked to below suggests that humans may actually be adapted away from strict monogamy.

Human sexual behavior resembles avian sexual behavior rather strongly. Both women and men have incentives to cheat, but for rather different reasons. Strict monogamy makes lower-status men better off at the cost of higher-status men and most women.

Wildly speculating here, but maybe the anxiety and disinterest men sometimes feel after their first time with a new partner is "meant" to remove them from the situation so the next guy can have a turn?

How does that impulse outcompete alternatives? If I feel a need to give the other guy a turn, and the other guy feels a need to monopolize his sexual partner, he will reproduce more than I will. Genes reproduce on the level of individuals, not societies.