The π orbitals of an alkene can stabilize a transition state by helping to delocalize the positive charge of the carbocation. For instance the unsaturated tosylate will react more quickly (10 times faster for aqueous solvolysis) with a nucleophile than the saturated tosylate.
The carbocationic intermediate will be stabilized by resonance where the positive charge is spread over several atoms. In the diagram below this is shown.
Here is a different view of the same intermediates.
Even if the alkene is more remote from the reacting center the alkene can still act in this way. For instance in the following alkyl benzenesulfonate the alkene is able to delocalise the carbocation.