We demonstrate pre-transition effects in space-time in trajectories of systems in which the dynamics displays a first-order phase transition between distinct dynamical phases. These effects are analogous to those observed for thermodynamic first-order phase transitions, most notably the hydrophobic effect in water. Considering the (infinite temperature) East model as an elementary example, we study the properties of "space-time solvation" by examining trajectories where finite space-time

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... are conditioned to be inactive in an otherwise active phase. Analogous to ideal hydrophobic solutes in water, we show that solvating an inactive region of space-time within an active trajectory shows two regimes in the dynamical equivalent of solvation free energy: an "entropic" small solute regime in which uncorrelated fluctuations are sufficient to evacuate activity from the solute, and an "energetic" large solute regime which involves the formation of a solute-induced inactive domain with an associated active-inactive interface bearing a dynamical interfacial tension. We also show that as a result of this dynamical interfacial tension there is a dynamical analog of the hydrophobic collapse that drives the assembly of large hydrophobes in water. We discuss the general relevance of these results to the properties of dynamical fluctuations in systems with slow collective relaxation such as glass formers.